A155756 A triangle sequence of polynomial coefficients: p(x,n)=(-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(m + 1)^ n*x^m, {m, 0, Infinity}]/(x^n); t(n,m)=coefficients(p(x,n))

1, 2, 1, 9, 33, 17, 1, 64, 610, 1180, 595, 70, 1, 625, 11315, 48135, 67245, 33309, 5463, 227, 1, 7776, 228531, 1708496, 4680256, 5339376, 2610776, 522256, 36996, 656, 1, 117649, 5104701, 59221547, 268424247, 551826072, 547629432, 265213752

Offset

0,2

Comments

Row sums are:

{1, 3, 60, 2520, 166320, 15135120, 1764322560, 251415964800, 42405492729600,

8269071082272000, 1831223377855872000,...}

Links

Table of n, a(n) for n=0..37.

Formula

p(x,n)=(-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(m + 1)^ n*x^m, {m, 0, Infinity}]/(x^n);

t(n,m)=coefficients(p(x,n))

Example

{1},
{2, 1},
{9, 33, 17, 1},
{64, 610, 1180, 595, 70, 1},
{625, 11315, 48135, 67245, 33309, 5463, 227, 1},
{7776, 228531, 1708496, 4680256, 5339376, 2610776, 522256, 36996, 656, 1},
{117649, 5104701, 59221547, 268424247, 551826072, 547629432, 265213752, 60598872, 5973447, 211067, 1773, 1},
{2097152, 126049540, 2096330664, 14207052574, 46045110980, 77141715750, 69121920120, 33230236215, 8354341710, 1033843525, 56182148, 1079835, 4586, 1},
{43046721, 3423831975, 77260093125, 731924512275, 3437765874135, 8726259755169, 12530549950875, 10383294890925, 4962943954925, 1341045544795, 196133991009, 14379274775, 462872275, 5125125, 11495, 1},
{1000000000, 101687123005, 2990094781780, 37741149307420, 242438838391372, 868344303959314, 1827360860004316, 2325557079854008, 1811458523454580, 861797333170774, 246383386597348, 40969332412372, 3750975005596, 173027847370, 3467306284, 23028328, 28132, 1},
{25937424601, 3282485876153, 121943147250045, 1978276502459037, 16660492641426414, 80063816942156718, 232316612953922574, 421187996783503950, 486314819595330726, 360267425295559782, 170739215087786982, 51078738006949926, 9404267410779150, 1022873614221774, 61697673399918, 1869498032814, 24180490437, 99233445, 67553, 1}

Mathematica

p[x_, n_] = (-1)^(n + 1)*(x - 1)^( 3*n + 1)*Sum[(Binomial[m, n]*Binomial[m + 1, n + 1]/(m - n + 1))*(m + 1)^n*x^m, {m, 0, Infinity}]/(x^n);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A320534 A012892 A013071 * A204432 A139004 A074951

Adjacent sequences: A155753 A155754 A155755 * A155757 A155758 A155759

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Jan 26 2009

Status

approved