# [seqfan] Re: any value storing constants of power ratios?

Richard Mathar mathar at strw.leidenuniv.nl
Sun Feb 8 21:50:46 CET 2009

```These are three constants made up by my own, plus 18 others from
a table of formulas, concerning summation over fractions.
Does somebody know the closed form solution to A000020, that would be
an erratum to the table? It is probably not far off the one I am quoting.

In most cases I did not check that the analytic formulas are close to
some partial sums of the series of A000004 to A000021, which some members
of the list are encouraged to add as a check.

%I A000001
%S A000001 2,0,6,5,8,8,6,5,3,8,8,8,4,1,3,5,2,5,0,9,0,3,1,4,2,2
%N A000001 Decimal expansion of sum_{n=2...infinity} 1/(n* (log n)^3).
%C A000001 Cubic analog of A115563. Evaluated by direct summation of the first 160
%C A000001 terms and accumulating the remainder with the 5 non-trivial terms in the Euler-Maclaurin expansion.
%e A000001 2.06588...
%Y A000001 Cf. A115563.
%K A000001 cons,nonn
%O A000001 1,1
%A A000001 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000002
%S A000002 2,5,5,9,1,1,9,7,4,2,9,8,6,7,3,1,4,1,8,5,7,2,0,2,0,9
%N A000002 Decimal expansion of sum_{n=2...infinity} 1/(n* (log n)^4).
%C A000002 Quartic analog of A115563. Evaluated by direct summation of the first 160
%C A000002 terms and accumulating the remainder with the 5 non-trivial terms in
the Euler-Maclaurin expansion.
%e A000002 2.5591197429867314185720209
%Y A000002 Cf. A115563.
%K A000002 cons,nonn
%O A000002 1,1
%A A000002 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000003
%S A000003 3,4,2,9,8,1,6,2,6,0,0,2,3,0,5,6,0,6,5,0,2,2,4,1,1,5,8
%N A000003 Decimal expansion of sum_{n=2...infinity} 1/(n* (log n)^5).
%C A000003 Quintic analog of A115563. Evaluated by direct summation of the first 160
%C A000003 terms and accumulating the remainder with the 5 non-trivial terms in the Euler-Maclaurin expansion.
%e A000003 3.42981626002305606502241158...
%Y A000003 Cf. A115563.
%K A000003 cons,nonn
%O A000003 1,1
%A A000003 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000004
%S A000004 9,7,0,8,0,3,8,8,4,3,0,0,7,7,5,8,4,7,3,2,9,7,7,4,4,9,8,1,8,8,2,
%T A000004 2,7,1,4,5,6,4,3,8,5,2,2,6,8,6,3,8,4,9,6,6,7,5,7,6,8,1,9,3,0,8,
%U A000004 9,6,7,5,2,0,5,8,2,4,7,8,1,6,5,4,2,8,3,5,1,9,2,4,6,9,1,4,3,4,1
%N A000004 Decimal expansion of sum_{n=0..infinity} (-1)^n/(2^(3n)*(3n+1)).
%D A000004 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.17
%F A000004 (A093602+A002391)/3 .
%e A000004 0.97080388430077584732977...
%p A000004 evalf((Pi/sqrt(3)+log(3))/3) ;
%K A000004 cons,easy,nonn
%O A000004 0,1
%A A000004 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000005
%S A000005 1,7,1,6,8,1,4,6,9,2,8,2,0,4,1,3,5,2,3,0,7,4,7,1,3,2,3,3,4,4,0,
%T A000005 9,9,4,2,3,1,6,0,5,6,6,1,2,0,1,2,4,9,7,8,8,3,5,8,0,5,0,1,1,0,8,
%U A000005 1,5,3,8,4,3,6,7,1,8,7,4,2,7,5,8,2,0,0,2,7,4,3,9,3,0,3,4,9,3,1
%N A000005 Decimal expansion of sum_{n=1..infinity} (-1)^(n-1)/(n^2-1/4)^2.
%D A000005 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.18
%F A000005 8 minus A019692.
%e A000005 1.7168146928204135...
%p A000005 evalf(8-2*Pi) ;
%K A000005 cons,easy,nonn
%O A000005 1,2
%A A000005 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000006
%S A000006 0,4,9,8,0,8,5,6,6,7,4,7,6,3,0,5,1,6,8,6,2,7,4,0,7,0,7,7,2,0,9,
%T A000006 9,1,3,3,1,1,8,7,7,1,5,0,4,6,0,1,1,0,0,9,2,2,0,8,3,7,0,6,7,4,9,
%U A000006 4,3,3,3,4,2,5,2,5,3,9,6,0,9,8,7,1,5,7,4,8,9,5,3,0,1,7,8,4,8,7
%N A000006 Decimal expansion of sum_{n=1..infinity} 1/(n*(25n^2-1)).
%D A000006 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.26
%e A000006 0.049808566747630516862...
%p A000006 evalf(5*log(5)/4-5/2+sqrt(5)/2*log((1+sqrt(5))/2)) ;
%K A000006 cons,easy,nonn
%O A000006 0,2
%A A000006 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000007
%S A000007 0,3,4,2,1,2,7,9,4,1,2,2,0,5,5,1,5,5,9,2,7,3,3,2,0,9,8,3,0,0,1,
%T A000007 4,1,6,9,3,1,2,2,2,3,6,1,0,5,4,5,4,6,3,4,6,8,5,8,4,3,4,0,1,5,1,
%U A000007 9,4,4,3,0,2,8,6,8,3,7,6,7,3,0,2,8,8,1,5,2,2,1,5,0,2,8,6,2,1,3
%N A000007 Decimal expansion of sum_{n=1..infinity} 1/(n*(36n^2-1)).
%D A000007 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.27
%F A000007 A016627+3*(A156057-1).
%e A000007 0.03421279412205515...
%p A000007 2*ln(2)-3+3/2*ln(3) ;
%K A000007 cons,easy,nonn
%O A000007 0,2
%A A000007 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000008
%S A000008 2,8,9,8,6,8,1,3,3,6,9,6,4,5,2,8,7,2,9,4,4,8,3,0,3,3,3,2,9,2,0,
%T A000008 5,0,3,7,8,4,3,7,8,9,9,8,0,2,4,1,3,5,9,6,8,7,5,4,7,1,1,1,6,4,5,
%U A000008 8,7,4,0,0,1,4,9,4,0,8,0,6,4,0,1,7,4,7,6,6,7,2,5,7,8,0,1,2,3,9
%N A000008 Decimal expansion of sum_{n=0..inf} (n!/(n+2)!)^2.
%D A000008 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.31
%F A000008 A002388/3-3 .
%e A000008 0.28986813369645287294483...
%p A000008 evalf(1/3*Pi^2-3) ;
%K A000008 cons,easy,nonn
%O A000008 0,1
%A A000008 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000009
%S A000009 0,2,9,9,0,1,1,0,0,2,7,2,3,3,9,6,5,4,7,0,8,6,2,2,7,4,9,9,6,9,0,
%T A000009 3,7,7,8,3,8,2,8,4,2,4,8,5,1,8,1,0,1,9,7,6,5,6,6,0,3,3,3,7,3,4,
%U A000009 4,0,5,5,0,1,1,2,0,5,6,0,4,8,0,1,3,1,0,7,5,0,4,4,3,3,5,0,9,2,9
%N A000009 Decimal expansion of sum_{n=0..inf} (n!/(n+3)!)^2.
%D A000009 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.32
%F A000009 A091476 - 2.4375 .
%e A000009 0.0299011002723396547...
%p A000009 1/4*Pi^2-39/16 ;
%K A000009 cons,easy,nonn
%O A000009 0,2
%A A000009 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000010
%S A000010 1,0,2,5,5,1,9,7,4,5,6,9,3,6,8,7,1,4,0,2,3,7,6,3,1,3,0,3,0,5,6,
%T A000010 8,6,2,2,9,2,9,1,3,6,2,6,4,9,9,2,3,7,0,9,6,2,3,0,2,2,7,9,5,3,9,
%U A000010 7,4,1,5,5,2,4,9,2,7,2,4,5,0,5,4,1,5,5,3,4,7,3,6,4,9,9,9,5,2,8
%N A000010 Decimal expansion of sum_{n=1..inf} 3^n*(n!)^2/(2n)!.
%D A000010 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.34
%F A000010 3+10*A019694*A020760.
%e A000010 10.255197456936871...
%p A000010 4/3*Pi*3^(1/2)+3 ;
%K A000010 cons,easy,nonn
%O A000010 2,3
%A A000010 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000011
%S A000011 1,0,6,9,7,3,3,1,9,2,0,5,2,0,4,8,4,1,1,2,4,3,1,2,8,5,0,1,6,9,8,
%T A000011 2,5,6,8,2,9,3,9,6,4,5,9,1,6,6,2,4,2,8,3,1,2,3,9,0,1,5,5,2,9,9,
%U A000011 8,5,6,4,1,8,0,5,1,5,1,3,6,1,4,1,1,9,7,4,1,5,2,0,2,7,7,7,5,1,5
%N A000011 Decimal expansion of sum_{n=1..inf} n*(n!)^2/(2n)!.
%D A000011 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.35
%F A000011 10*A021139*(9+A000796*A002194).
%e A000011 1.069733192052..
%p A000011 2/3+2/27*Pi*3^(1/2) ;
%K A000011 cons,easy,nonn
%O A000011 1,3
%A A000011 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000012
%S A000012 2,0,0,5,1,1,0,8,7,5,6,4,2,3,0,2,9,0,7,6,2,7,4,3,6,3,9,1,7,1,9,
%T A000012 3,1,6,9,3,7,8,8,2,9,8,7,4,9,9,2,9,3,6,0,7,6,2,0,5,8,1,4,3,8,8,
%U A000012 6,4,9,5,8,5,6,4,1,4,1,1,5,7,9,0,8,8,4,5,8,0,8,9,3,5,1,8,0,8,1
%N A000012 Decimal expansion of sum_{n=1..inf} n^2*(n!)^2/(2n)!.
%D A000012 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.36
%F A000012 (2/81)*(54+5*Pi*sqrt(3)).
%e A000012 2.00511087564...
%p A000012 4/3+10/81*Pi*3^(1/2) ;
%K A000012 cons,easy,nonn
%O A000012 1,1
%A A000012 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000013
%S A000013 4,9,9,0,3,8,4,6,0,4,3,6,2,1,2,4,9,4,9,9,2,5,4,5,4,2,1,0,6,8,5,
%T A000013 4,2,6,2,2,4,5,5,5,8,1,3,6,0,9,3,6,8,6,7,6,5,7,5,2,1,1,9,9,3,6,
%U A000013 4,4,6,7,5,5,6,9,3,2,5,9,6,7,2,8,6,2,6,4,1,0,6,4,8,4,5,5,7,1,3
%N A000013 Decimal expansion of sum_{n=1..inf} n^3/binomial(2n,n).
%D A000013 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.37
%F A000013 (2/243)*(405+37*Pi*sqrt(3)).
%e A000013 4.990384604362124...
%p A000013 10/3+74/243*Pi*3^(1/2);
%K A000013 cons,easy,nonn
%O A000013 1,1
%A A000013 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000014
%S A000014 5,8,6,7,8,1,9,9,8,7,6,6,9,8,2,1,1,5,8,4,3,6,9,8,0,8,4,9,6,0,1,
%T A000014 3,5,2,7,4,7,3,3,8,7,5,9,1,0,3,1,5,7,2,5,4,7,5,6,7,3,5,2,3,5,5,
%U A000014 6,7,5,3,3,5,1,7,0,7,5,5,1,6,3,6,9,1,7,6,7,6,1,5,2,7,8,8,2,6,6
%N A000014 Decimal expansion of sum_{n=1..inf} (-1)^(n-1)*2^n/binomial(2n,n).
%D A000014 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.38
%F A000014 (1+log(2+sqrt(3))/sqrt(3))/3 .
%e A000014 0.58678199876...
%p A000014 1/3+1/9*ln(2+3^(1/2))*3^(1/2) ;
%K A000014 cons,easy,nonn
%O A000014 0,1
%A A000014 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000015
%S A000015 2,7,4,4,3,2,7,1,5,2,7,7,1,2,0,3,2,3,1,1,1,1,5,4,6,5,8,6,3,6,0,
%T A000015 4,8,4,3,4,0,3,3,9,6,5,6,5,4,6,0,3,2,2,3,1,7,2,3,8,0,5,6,0,4,8,
%U A000015 8,3,1,9,4,0,4,8,9,7,2,3,6,8,9,0,5,5,6,9,0,8,9,1,9,2,2,1,1,7,5
%N A000015 Decimal expansion of sum_{n=1..inf} (-1)^(n-1)*n/binomial(2n,n).
%D A000015 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.39
%F A000015 2*(15+2*A002163*A002390)/125
%e A000015 0.274432715277120323..
%p A000015 6/25+4/125*5^(1/2)*ln(1/2+1/2*5^(1/2)) ;
%K A000015 cons,easy,nonn
%O A000015 0,1
%A A000015 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000016
%S A000016 1,2,5,5,6,7,2,8,4,7,2,2,8,7,9,6,7,6,8,8,8,8,4,5,3,4,1,3,6,3,9,
%T A000016 5,1,5,6,5,9,6,6,0,3,4,3,4,5,3,9,6,7,7,6,8,2,7,6,1,9,4,3,9,5,1,
%U A000016 1,6,8,0,5,9,5,1,0,2,7,6,3,1,0,9,4,4,3,0,9,1,0,8,0,7,7,8,8,2,4
%N A000016 Decimal expansion of sum_{n=1..inf} (-1)^(n-1)*n^2/binomial(2n,n).
%C A000016 The numerator in the Apelblat table lacks the square (typo).
%D A000016 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.40.
%F A000016 4*(5-A002164*A002390)/125 .
%e A000016 0.125567284722879676...
%p A000016 4/25-4/125*5^(1/2)*ln(1/2+1/2*5^(1/2)) ;
%K A000016 cons,easy,nonn
%O A000016 0,2
%A A000016 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000017
%S A000017 0,8,4,1,7,7,4,0,8,0,0,0,8,3,3,2,0,3,0,3,5,5,4,8,6,9,5,3,8,4,6,
%T A000017 6,7,2,6,7,8,8,5,5,3,1,8,4,0,3,9,9,8,8,4,5,8,2,8,8,7,7,5,9,0,1,
%U A000017 1,7,7,4,1,6,8,9,0,6,6,6,5,1,8,7,0,6,4,8,1,0,6,4,0,3,2,2,6,9,1
%N A000017 Decimal expansion of sum_{n=1..inf} (-1)^(n-1)/(n*binomial(2n,n)).
%D A000017 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.42.
%F A000017 log(A014176/2)*A020762.
%e A000017 0.084177408000833..
%p A000017 1/5*ln(1/2+1/2*2^(1/2))*5^(1/2) ;
%Y A000017 Cf. A073010.
%K A000017 cons,easy,nonn
%O A000017 0,2
%A A000017 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000018
%S A000018 9,5,0,2,3,9,6,0,5,1,1,6,6,4,3,2,5,8,9,8,1,6,2,7,9,5,2,9,5,1,4,
%T A000018 2,6,9,0,9,1,6,9,7,3,0,8,5,1,0,5,8,9,0,1,8,2,5,2,8,9,6,5,4,5,4,
%U A000018 3,3,0,0,6,2,1,4,3,3,7,0,2,3,1,5,4,3,4,8,7,8,4,6,5,2,5,9,3,6,0
%N A000018 Decimal expansion of sum_{n=0..inf} (-1)^n/((2n+1)^2*binomial(2n,n)).
%D A000018 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.45.
%F A000018 A013661-3*A002390^2.
%e A000018 0.9502396051166432589..
%p A000018 1/6*Pi^2-3*ln(1/2+1/2*5^(1/2))^2 ;
%K A000018 cons,easy,nonn
%O A000018 0,3
%A A000018 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000019
%S A000019 4,8,0,8,2,2,7,6,1,2,6,3,8,3,7,7,1,4,1,5,9,8,9,5,2,6,4,6,0,4,5,
%T A000019 7,9,9,9,6,3,0,5,9,9,4,5,1,6,9,3,6,1,9,9,5,5,2,7,1,6,9,0,8,6,2,
%U A000019 2,1,3,6,7,3,5,2,8,2,3,1,4,5,2,5,2,3,6,0,7,4,5,8,2,3,4,9,4,4,3
%N A000019 Decimal expansion of sum_{n=1..inf} (-1)^(n-1)/(n^3*binomial(2n,n)).
%D A000019 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.46.
%F A000019 2*A002117/5 .
%e A000019 0.480822761263837714..
%p A000019 2/5*Zeta(3) ;
%K A000019 cons,easy,nonn
%O A000019 0,1
%A A000019 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000020
%S A000020 5,2,2,9,4,6,1,9,2,1,3,3,3,3,5,1,0,8,4,9,1,1,8,5,1,8,3,5,2,7,3,
%T A000020 0,3,5,4,0,1,6,3,0,4,4,5,9,1,7,4,3,9,7,7,8,4,1,4,6,5,9,4,1,0,1,
%U A000020 4,1,4,4,2,0,7,3,5,7,7,6,4,4,1,3,2,9,9,3,1,5,0,4,2,6,2,1,9,1,3
%N A000020 Decimal expansion of sum_{n=1..inf} 1/(n^3*binomial(2n,n)).
%C A000020 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.47 gives Pi*sqrt(3)*(psi(2/3)-psi(1/3))/72-Zeta(3)/3 which is negative and therefore not correct.
%e A000020 0.522946...
%K A000020 cons,easy,nonn
%O A000020 0,1
%A A000020 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

%I A000021
%S A000021 1,1,1,5,3,5,5,0,7,1,6,5,0,4,1,0,5,4,0,7,6,7,0,5,8,3,7,4,5,5,5,
%T A000021 8,3,0,9,3,7,9,4,5,8,2,7,1,8,4,4,6,4,5,8,5,7,2,4,6,6,0,4,5,5,2,
%U A000021 9,6,8,7,0,5,2,6,3,0,2,1,4,0,6,0,6,0,2,3,8,4,8,5,0,3,6,7,2,6,8
%N A000021 Decimal expansion of sum_{n=0..inf} binomial(4n,n)/2^(6n).
%D A000021 Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.49.
%F A000021 (1+A020760)*A010503 .
%e A000021 1.1153550716504105..
%p A000021 1/2*(1+1/3*3^(1/2))*2^(1/2);
%K A000021 cons,easy,nonn
%O A000021 1,4
%A A000021 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 08 2009

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