will be traveling for 2 weeks
N. J. A. Sloane
njas at research.att.com
Thu Jan 3 02:55:26 CET 2008
starting Friday, i will be away for 2+ weeks.
Best regards
Neil
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To: seqfan at ext.jussieu.fr
Date: Wed, 2 Jan 2008 21:50:02 -0500
Subject: Re: G.f. for C(q^n,n)? - Generalized
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Seqfans,
The identity can be generalized further.
Let F(x) be any formal power series in x such that F(0)=1.
Then
(5) Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b)
where [y^n] G(y) denotes the coefficient of y^n in G(y).
If we let F(x) = exp(x), then we have the nontrivial result
(6) Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! =
Sum_{n>=0} (m*q^n + b)^n * x^n / n!
which becomes trivial at b=0.
Example of (6): q=2, m=1, b=1:
exp(x) + 2*exp(2x) + 2^4*exp(4x)*x^2/2! + 2^9*exp(8x)*x^3/3! +...+
2^(n^2)*exp(2^n*x)/n! +...
= 1 + 3x + 5^2*x^2/2! + 9^3*x^3/3! + 17^4*x^4/4! +...+
(2^n+1)^n*x^n/n! +...
Example of (6): q=2, m=1, b=-1:
exp(-x) + 2*exp(-2x) + 2^4*exp(-4x)*x^2/2! + 2^9*exp(-8x)*x^3/3! +...+
2^(n^2)*exp(-2^n*x)/n! +...
= 1 + x + 3^2*x^2/2! + 7^3*x^3/3! + 15^4*x^4/4! +...+
(2^n-1)^n*x^n/n! +...
Example of (5): F(x) = 1+x+x^2, q=2, m=1, b=0:
Sum_{n>=0} log( 1 + 2^n*x + 2^(2n)*x^2 )^n / n! =
Sum_{n>=0} T(2^n, n) * x^n
where T(2^n,n) = trinomial coefficient of x^n in (1+x+x^2)^(2^n).
And many other examples can be given.
So the question arises, does (5) have useful applications?
Does it offer a g.f. for some significant sequences in OEIS?
Paul
On Sun, 30 Dec 2007 21:03:39 -0500 pauldhanna at juno.com writes:
Seqfans,
Recall the identity:
(2) Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} C(q^n,n)*x^n.
From this, I found the more general statements:
(3) Sum_{n>=0} m^n*log(1 + q^n*x)^n/n! = Sum_{n>=0} C(m*q^n,n)*x^n.
(4) Sum_{n>=0} m^n * (1 + q^n*x)^b * log(1 + q^n*x)^n/n! =
Sum_{n>=0} C(m*q^n + b, n)*x^n.
Identity (4) is very interesting ... I wonder if it leads to other
results?
It certainly can lead to many significant sequences!
What I would really like is for formula (4) to allow the g.f.
A(x,m,b) = Sum_{n>=0} C(m*q^n + b, n)*x^n
to be manipulated to solve some functional equation ...
Any ideas along these lines from anyone?
Paul
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<DIV>Seqfans,</DIV>
<DIV>
<DIV> The identity can be generalized further.=
=20
</DIV>
<DIV>Let F(x) be any formal power series in x such that F(0)=3D1. </DIV>
<DIV>Then </DIV>
<DIV>
<DIV>(5) Sum_{n>=3D0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! =
=3D =20
</DIV>
<DIV> Sum_{n>=3D0} x^n * [y^n] F(y)^(m*q^n&=
nbsp;+=20
b) </DIV>
<DIV> </DIV>
<DIV>where [y^n] G(y) denotes the coefficient of y^n in G(y).</DIV></DIV>
<DIV> </DIV>
<DIV>If we let F(x) =3D exp(x), then we have the nontrivial result </D=
IV>
<DIV> </DIV>
<DIV>
<DIV>(6) Sum_{n>=3D0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! =3D&=
nbsp;=20
</DIV>
<DIV> Sum_{n>=3D0} (m*q^n + b)^n * x^n=
/=20
n! </DIV>
<DIV> </DIV>
<DIV>which becomes trivial at b=3D0. </DIV></DIV>
<DIV> </DIV>
<DIV>Example of (6): q=3D2, m=3D1, b=3D1:</DIV>
<DIV>exp(x) + 2*exp(2x) + 2^4*exp(4x)*x^2/2! + 2^9*exp(8x)*x^3/3! +...+</DI=
V>
<DIV> 2^(n^2)*exp(2^n*x)/n! +...</DIV>
<DIV>
<DIV>=3D 1 + 3x + 5^2*x^2/2! + 9^3*x^3/3! + 17^4*x^4/4! +...+ </DIV>
<DIV> (2^n+1)^n*x^n/n! +... </DIV></DIV>
<DIV> </DIV>
<DIV>
<DIV>Example of (6): q=3D2, m=3D1, b=3D-1:</DIV>
<DIV>exp(-x) + 2*exp(-2x) + 2^4*exp(-4x)*x^2/2! + 2^9*exp(-8x)*x^3/3!=20
+...+</DIV>
<DIV> 2^(n^2)*exp(-2^n*x)/n! +...</DIV>
<DIV>
<DIV>=3D 1 + x + 3^2*x^2/2! + 7^3*x^3/3! + 15^4*x^4/4! +...+ </DIV>
<DIV> (2^n-1)^n*x^n/n! +...</DIV></DIV>
<DIV> </DIV></DIV>
<DIV>Example of (5): F(x) =3D 1+x+x^2, q=3D2, m=3D1, b=3D0: </DIV>
<DIV>Sum_{n>=3D0} log( 1 + 2^n*x + 2^(2n)*x^2 )^n / n! =3D =
=20
<DIV> Sum_{n>=3D0} T(2^n, n) * x^n =20
</DIV></DIV></DIV>
<DIV>where T(2^n,n) =3D trinomial coefficient of x^n in (1+x+x^2)^(2^n).=20
</DIV>
<DIV> </DIV>
<DIV>And many other examples can be given. </DIV>
<DIV> </DIV>
<DIV>So the question arises, does (5) have useful applicatio=
ns?=20
</DIV>
<DIV>Does it offer a g.f. for some significant sequences in OEIS? </DI=
V>
<DIV> Paul </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>On Sun, 30 Dec 2007 21:03:39 -0500 <A=20
href=3D"mailto:pauldhanna at juno.com">pauldhanna at juno.com</A> writes:</DIV>
<BLOCKQUOTE dir=3Dltr=20
style=3D"PADDING-LEFT: 10px; MARGIN-LEFT: 10px; BORDER-LEFT: #000000 2px so=
lid">
<DIV>Seqfans, </DIV>
<DIV> Recall the identity: </DIV>
<DIV>(2) Sum_{n>=3D0} log(1 + q^n*x)^n/n! =3D Sum_{n>=
=3D0}=20
C(q^n,n)*x^n. </DIV>
<DIV> </DIV>
<DIV>From this, I found the more general statements: </DIV>
<DIV> </DIV>
<DIV>(3) Sum_{n>=3D0} m^n*log(1 + q^n*x)^n/n! =3D Sum_{n&g=
t;=3D0}=20
C(m*q^n,n)*x^n.</DIV>
<DIV> </DIV>
<DIV>
<DIV>(4) Sum_{n>=3D0} m^n * (1 + q^n*x)^b * log(1 + q^n*x)^n/n! =
=3D =20
</DIV>
<DIV> Sum_{n>=3D0} C(m*q^n + b, n)*x^n.</=
DIV>
<DIV> </DIV>
<DIV>Identity (4) is very interesting ... I wonder if it leads to other=
=20
results? </DIV>
<DIV>It certainly can lead to many significant sequences! </DIV>
<DIV> </DIV>
<DIV>What I would really like is for formula (4) to allow the g=
.f.=20
</DIV>
<DIV> A(x,m,b) =3D Sum_{n>=3D0} C(m*q^n + b, n)=
*x^n=20
</DIV>
<DIV>to be manipulated to solve some functional equation ... </DIV>
<DIV> </DIV>
<DIV>Any ideas along these lines from anyone?</DIV>
<DIV> Paul</DIV></DIV></BLOCKQUOTE></BODY><=
/HTML>
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