Stirling 1st problem in Infinite sums

• To: mathgroup at smc.vnet.net
• Subject: [mg96530] Stirling 1st problem in Infinite sums
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Mon, 16 Feb 2009 06:53:28 -0500 (EST)

```The problem is that I can't get Mathematica to give values for the
Stirling 1st Infinite sums.
Daniel Lichenbau said I should apply here.
A lot of new infinite sum polynomial have been found recently.
http://www.research.att.com/~njas/sequences/?q=bagula+infinite+sum&sort=0&fmt=0&language=english&go=Search

These new infinite sum polynomials below  come from noticing that
(k+1)^n-> Eulerian
(2*k+1)^n-> MacMahon
Binomial[k,n]--> like the Hanna type
were all good Infinite sums of the general
1/(1-x)^n=Sum[f[k,n]*x^k,{k,0,Infinity}]
I said to myself that the combinations:
(k+1)^n*Binomial[k,n]
(2*k+1)^n*Binomial[k,n]
were also of that type.
I was hoping for something like Eulerian or a MacMahon,
but got some sort of higher Stirling like sequence.
From my work on q-factorials and q-combinations ,
the
StirlingS1[k,n]
level is probably fundamental as an Infinite sum as well,
but it won't calculate in Mathematica:
Clear[p, x, n, m]
p[x_, n_] = (1 - x)^(n + 1)*Sum[(-1)^(k + n)*StirlingS1[k, n]*x^k, {k,
0, Infinity}]
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}]
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]
Flatten[%]

%I A156653
%S A156653
1,1,3,1,16,13,1,125,171,39,1,1296,2551,1091,101,1,16807,43653,28838,
%T A156653
5498,243,1,262144,850809,780585,243790,24270,561,1,4782969,18689527,
%U A156653
22278189,10073955,1733035,98661,1263,1,100000000,457947691,677785807
%N A156653 Coefficients of a higher level infinite sum polynomial:
p(x,n)=(1 - x)^(2n + 1)/((n + 1)*x^n)*Sum[(k + 1)^n*Binomial[k, n]*x^ k,
{k, 0, Infinity}]. %C A156653 Roe sums are:A001761;
%C A156653 {1, 1, 4, 30, 336, 5040, 95040, 2162160, 57657600,
1764322560, 60949324800,...}. %F A156653 p(x,n)=(1 - x)^(2n + 1)/((n +
1)*x^n)*Sum[(k + 1)^n*Binomial[k, n]*x^ k, {k, 0, Infinity}];
%F A156653 t(n,m)=coefficients(p(x,n)). %e A156653 {1}, %e A156653 {1},
%e A156653 {3, 1}, %e A156653 {16, 13, 1}, %e A156653 {125, 171, 39, 1},
%e A156653 {1296, 2551, 1091, 101, 1}, %e A156653 {16807, 43653, 28838,
5498, 243, 1}, %e A156653 {262144, 850809, 780585, 243790, 24270, 561,
1}, %e A156653 {4782969, 18689527, 22278189, 10073955, 1733035, 98661,
1263, 1}, %e A156653 {100000000, 457947691, 677785807, 410994583,
106215619, 10996369, 379693, 2797, 1}, %e A156653 {2357947691,
12400462713, 22055317500, 17027114412, 6066172434, 976428894, 64468572,
1406460, 6123, 1} %t A156653 Clear[p, x, n, m];
%t A156653 p[x_, n_] = (1 - x)^( 2n + 1)/((n + 1)*x^n)*Sum[(k +
1)^n*Binomial[k, n]*x^k, {k, 0, Infinity}];
%t A156653 Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
%t A156653 Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x],
{n, 0, 10}];
%t A156653 Flatten[%] %K A156653 nonn,tabl
%O A156653 0,3
%A A156653 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 12 2009

%I A156654
%S A156654
1,3,1,25,22,1,343,515,101,1,6561,14156,5766,396,1,161051,456197,299342,
%T A156654
49642,1447,1,4826809,16985858,15796159,4592764,371239,5090,1,170859375,
%U A156654
719818759,878976219,383355555,58474285,2550165,17481,1,6975757441
%N A156654 Coefficients of a higher level infinite sum polynomial:
p(x,n)=(1 - x)^(2n + 1)/(x^n)*Sum[(2*k + 1)^n*Binomial[k, n]*x^k, {k, 0,
Infinity}]. %C A156654 Roe sums are:A052714;
%C A156654 {1, 4, 48, 960, 26880, 967680, 42577920, 2214051840,
132843110400, %C A156654 9033331507200, 686533194547200 %F A156654
p(x,n)=(1 - x)^(2n + 1)/(x^n)*Sum[(2*k + 1)^n*Binomial[k, n]*x^k, {k, 0,
Infinity}];
%F A156654 t(n,m)=coefficients(p(x,n)). %e A156654 {1}, %e A156654 {3,
1}, %e A156654 {25, 22, 1}, %e A156654 {343, 515, 101, 1}, %e A156654
{6561, 14156, 5766, 396, 1}, %e A156654 {161051, 456197, 299342, 49642,
1447, 1}, %e A156654 {4826809, 16985858, 15796159, 4592764, 371239,
5090, 1}, %e A156654 {170859375, 719818759, 878976219, 383355555,
58474285, 2550165, 17481, 1}, %e A156654 {6975757441, 34264190872,
52246537948, 31191262504, 7488334150, 660394024, 16574428, 59032, 1}, %e
A156654 {322687697779, 1811734208009, 3329783850284, 2563367714324,
872277734234, 126505988606, 6870434876, 103682276, 196811, 1}, %e
A156654 {16679880978201, 105414122807918, 227501403350541,
216602727685224, 97632310949922, 20706515546388, 1928212521522,
67389166824, 630891141, 649518, 1} %t A156654 Clear[p, x, n, m];
%t A156654 p[x_, n_] = (1 - x)^(2n + 1)/(x^n)*Sum[(2*k +
1)^n*Binomial[k, n]*x^k, {k, 0, Infinity}];
%t A156654 Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
%t A156654 Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x],
{n, 0, 10}];
%t A156654 Flatten[%] %K A156654 nonn,tabl
%O A156654 0,2
%A A156654 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 12 2009

--
Respectfully, Roger L. Bagula