# [seqfan] Re: A173279 and A173280

Richard Mathar mathar at strw.leidenuniv.nl
Mon Feb 22 17:00:13 CET 2010

```In response to http://list.seqfan.eu/pipermail/seqfan/2010-February/003716.html
this is roughly what I would come up with:

%I A173279
%S A173279 1,1,2,1,6,1,24,2,1,120,6,1,720,24,2,1,5040,120,6,1,40320,720,24,2,1,362880,
%T A173279 5040,120,6,1,3628800,40320,720,24,2,1,39916800,362880,5040,120,6,1,479001600,
%U A173279 3628800,40320,720,24,2,1,6227020800,39916800,362880,5040,120,6,1,87178291200
%N A173279 Irregular triangle M(n,k) read by rows: factorial of n-2*k, k=0..floor(n/2).
%C A173279 In the limit of j-> infinity, the power M^j approaches a limit described in A173280.
%F A173279 M(n,k) = A000142(n-2*k), k=0..[n/2], n>=0.
%F A173279 Row sums: sum_{k=0..n/2} M(n,k) = A136580(n).
%e A173279 Triangle starts in row n=0 as:
%e A173279 1;
%e A173279 1;
%e A173279 2, 1;
%e A173279 6, 1;
%e A173279 24, 2, 1;
%e A173279 120, 6, 1;
%e A173279 720, 24, 2, 1;
%e A173279 5040, 120, 6, 1;
%e A173279 40320, 720, 24, 2, 1;
%e A173279 362880, 5040, 120, 6, 1;
%e A173279 3628800, 40320, 720, 24, 2, 1;
%e A173279 39916800, 362880, 5040, 120, 6, 1;
%e A173279 479001600, 3628800, 40320, 720, 24, 2, 1;
%e A173279 ...
%p A173279 A173279 := proc(n,k) factorial(n-2*k) ; end proc: seq(seq(A173279(n,k),k=0..floor(n/2)),n=0..20) ; # R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 22 2010
%Y A173279 Cf. A000142, A136580, A173280
%K A173279 nonn,tabf,new
%O A173279 0,3
%A A173279 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 14 2010
%E A173279 keyword tabl replaced by tabf, R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 22 2010

%I A173280
%S A173280 1,1,3,7,29,129,757,5185,41155,368351,3671635,40295943,482758111,6268066531,
%T A173280 87668492115,1314023850727,21011431917453,357014074280785,6423561495057421,
%U A173280 122004755658629081,2439367774898883497,51213663674167659301,1126452985959434543237
%N A173280 First column of the matrix power A173279(.,.)^j in the limit j->infinity.
%C A173280 An infinite set of generalized triangles shifted down n times would have
%C A173280 constants tending to e using the analogous summation operation.
%F A173280 A000142(x) = A(x)/A(x^2), where A(x) and A000142(x) are the o.g.f.'s associated with A000142 and this sequence here.
%F A173280 sum_{n>=0} 1/a(n) = 2.519966353393413186683398448854995831308...
%F A173280 a(n) = (A173279^j)(n,0).
%F A173280 a(n) = sum_{t=0..n, n-t even} t!*a_{(n-t)/2}. [R. J. Mathar, Feb 22 2010]
%p A173280 A173280 := proc(n) option remember; local a,l; if n = 0 then 1; else a :=0 ; for l from n to 0 by -2 do a := a+ l!*procname((n-l)/2) : end do ; a ; end if; end proc:
%p A173280 seq(A173280(n),n=0..60) ; # R. J. Mathar, Feb 22 2010
%Y A173280 Cf. A000142
%K A173280 nonn,new
%O A173280 0,3
%A A173280 Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 14 2010
%E A173280 Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 22 2010

```