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A155948 A triangle of polynomial coefficients: q(x,n)=(-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(2*m + 1)^n*x^ m, {m, 0, Infinity}]/(x^n); p(x,n)=q(x,n)+x^n*q(1/x,n). 0
2, 6, 6, 34, 206, 206, 34, 370, 4840, 14950, 14950, 4840, 370, 6642, 142644, 792216, 1719618, 1719618, 792216, 142644, 6642, 161294, 5282074, 45682504, 158295424, 274902544, 274902544, 158295424, 45682504, 5282074, 161294, 4827538 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Row sums are:

{2, 12, 480, 40320, 5322240, 968647680, 225833287680, 64362486988800,

21711612277555200, 8467528788246528000, 3750345477848825856000,...}.

LINKS

Table of n, a(n) for n=0..31.

FORMULA

q(x,n)=(-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(2*m + 1)^n*x^ m, {m, 0, Infinity}]/(x^n);

p(x,n)=q(x,n)+x^n*q(1/x,n);

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

EXAMPLE

{2},

{6, 6},

{34, 206, 206, 34},

{370, 4840, 14950, 14950, 4840, 370},

{6642, 142644, 792216, 1719618, 1719618, 792216, 142644, 6642},

{161294, 5282074, 45682504, 158295424, 274902544, 274902544, 158295424, 45682504, 5282074, 161294},

{4827538, 227651778, 2907137246, 14984780406, 38115062856, 56677184016, 56677184016, 38115062856, 14984780406, 2907137246, 227651778, 4827538},

{170861562, 11016050364, 197554369086, 1459983311028, 5313520312650, 10834039248120, 14364959341590, 14364959341590, 10834039248120, 5313520312650, 1459983311028, 197554369086, 11016050364, 170861562},

{6975764002, 589883814920, 14194396260000, 144086584363300, 732524911117760, 2067582128226648, 3567123431959120, 4329697827271850, 4329697827271850, 3567123431959120, 2067582128226648, 732524911117760, 144086584363300, 14194396260000, 589883814920, 6975764002},

{322687717462, 34650253894894, 1078225225888096, 14442860573483848, 98895486888500764, 381079250530358380, 884703838750948216, 1338789895455044032, 1514739863757428308, 1514739863757428308, 1338789895455044032, 884703838750948216, 381079250530358380, 98895486888500764, 14442860573483848, 1078225225888096, 34650253894894, 322687717462},

{16679881037250, 2217074072477334, 86648480181566430, 1481836294429602306, 13190542163684355876, 67274063665962266004, 208944364733797179732, 415175601761715156180, 565159740290092550148, 603857707780596736740, 603857707780596736740, 565159740290092550148, 415175601761715156180, 208944364733797179732, 67274063665962266004, 13190542163684355876, 1481836294429602306, 86648480181566430, 2217074072477334, 16679881037250}

MATHEMATICA

Clear[p, x, n, m];

p[x_, n_] = (-1)^(n + 1)*(x - 1)^(3*n + 1)*Sum[(Binomial[m, n]* Binomial[m + 1, n + 1]/(m - n + 1))*(2*m + 1)^n*x^ m, {m, 0, Infinity}]/(x^n);

Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]

+ Reverse[ CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 0, 10}];

Flatten[%]

CROSSREFS

Sequence in context: A056603 A019198 A155164 * A228955 A328584 A226707

Adjacent sequences:  A155945 A155946 A155947 * A155949 A155950 A155951

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Jan 31 2009

STATUS

approved

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Last modified December 3 18:19 EST 2021. Contains 349467 sequences. (Running on oeis4.)