       Re: generalization of Bernoulli numbers to Pascal types

• To: mathgroup at smc.vnet.net
• Subject: [mg95131] Re: [mg95087] generalization of Bernoulli numbers to Pascal types
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Wed, 7 Jan 2009 04:11:16 -0500 (EST)

```You need to use a pattern in the definition of p

p[x_] = 1/Sum[x^(n - 1)/(2*n - 1)!!, {n, 1, Infinity}]

(Sqrt[2/Pi]*Sqrt[x])/
(E^(x/2)*Erf[Sqrt[x]/Sqrt])

You don't need to use both Series and SeriesCoefficient (I only show a
few terms)

m = 5;

Table[SeriesCoefficient[p[x], {x, 0, n}], {n, 0, m}]

{1, -(1/3), 2/45, -(2/945),
-(2/14175), 2/93555}

Or more simply,

CoefficientList[Series[p[x], {x, 0, m}], x]

{1, -(1/3), 2/45, -(2/945),
-(2/14175), 2/93555}

Bob Hanlon

On Tue, Jan 6, 2009 at 7:42 AM , Roger Bagula wrote:

> 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
>  11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814
> :http://www.geocities.com/rlbagulatftn/Index.html
> alternative email: rlbagula at sbcglobal.net

```

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