generalization of Bernoulli numbers to Pascal types

• To: mathgroup at smc.vnet.net
• Subject: [mg95087] generalization of Bernoulli numbers to Pascal types
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Tue, 6 Jan 2009 04:10:04 -0500 (EST)

```Here is a new way to use the exponential like
expansions to get a generalized Bernoulli number B(n,m):
( I thought of these about a week ago, but just got around to getting
Mathematica to give me an expansion)
I tried the (2*n-1)!! version but Mathematica failed on it.
Here is the code if anyone can get it to work:
Clear[p]
Table[(2*n - 1)!!, {n, 0, 10}]
p[x] = FullSimplify[1/Sum[x^(n - 1)/(2*n - 1)!!, {n, 1, Infinity}]]
Table[ SeriesCoefficient[
Series[p[x], {x, 0, 30}], n], {n, 0, 30}]

%I A154242
%S A154242 1,3,45,1890,56700,748440,10216206000,8756748000,2841962760000,
%T A154242 24946749107280000,8232427205402400000,103279541304139200000,
%U A154242 3101484625363300176000000,1431454442475369312000000
%N A154242 Bernoulli like expansion of polynomials: 1/Sum[x^(n -
1)/((2*n)!/n!), {n, 1, Infinity}]=2* Exp[-x/4]
*Sqrt[x]/(Sqrt[Pi]*Erf[Sqrt[x]/2]). denominators of the rational sequence.
%C A154242 The row sum sequences of the Pascal types are:
%C A154242 row(n,m)={2^n,n!,2^n*n!,(2*n-1)!!,(2*n)!/n!}. Each one can be
expanded as a Bernoulli like:
%C A154242
f(x)=x/(Sum[x^n/Row(n,m),{n,0,Infinity}]-1)=Sum[B[n,m]*x^n/n!,{n,0,Infinity}];
%C A154242 Here the m=4.
%F A154242 a(n)=Denominator[Expansion[1/Sum[x^(n - 1)/((2*n)!/n!), {n,
1, Infinity}]=2* Exp[-x/4] *Sqrt[x]/(Sqrt[Pi]*Erf[Sqrt[x]/2])]].
%t A154242 Clear[p]; p[x] = FullSimplify[1/Sum[x^(n - 1)/((2*n)!/n!),
{n, 1, Infinity}]];
%t A154242 Table[ Denominator[SeriesCoefficient[Series[p[x], {x, 0,
30}], n]], {n, 0, 30}]
%K A154242 nonn
%O A154242 0,2
%A A154242 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 05 2009

%I A154243
%S A154243 2,1,1,1,1,1,23,23,157,97051,1614583,331691,1418383997,5720927,
%T A154243 1868325937,1207461869239,118209298450003,3069893653,
%U A154243 14303719087308533,65108016166881997,310766859240153209819
%V A154243
2,-1,1,-1,-1,1,23,-23,157,97051,-1614583,-331691,1418383997,-5720927,
%W A154243 -1868325937,1207461869239,118209298450003,-3069893653,
%X A154243 -14303719087308533,65108016166881997,-310766859240153209819
%N A154243 Bernoulli like expansion of polynomials: 1/Sum[x^(n -
1)/((2*n)!/n!), {n, 1, Infinity}]=2* Exp[-x/4]
*Sqrt[x]/(Sqrt[Pi]*Erf[Sqrt[x]/2]). numerators of the rational sequence.
%C A154243 The row sum sequences of the Pascal types are:
%C A154243 row(n,m)={2^n,n!,2^n*n!,(2*n-1)!!,(2*n)!/n!}. Each one can be
expanded as a Bernoulli like:
%C A154243
f(x)=x/(Sum[x^n/Row(n,m),{n,0,Infinity}]-1)=Sum[B[n,m]*x^n/n!,{n,0,Infinity}];
%C A154243 Here the m=4.
%F A154243 a(n)=Numerator[Expansion[1/Sum[x^(n - 1)/((2*n)!/n!), {n, 1,
Infinity}]-&gt;2* Exp[-x/4] *Sqrt[x]/(Sqrt[Pi]*Erf[Sqrt[x]/2])]].
%t A154243 Clear[p]; p[x] = FullSimplify[1/Sum[x^(n - 1)/((2*n)!/n!),
{n, 1, Infinity}]];
%t A154243 Table[ Numerator[SeriesCoefficient[Series[p[x], {x, 0, 30}],
n]], {n, 0, 30}]
%K A154243 sign
%O A154243 0,1
%A A154243 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 05 2009

--
Respectfully, Roger L. Bagula