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A154336
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A triangular sequence of coefficients of polynomials: p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).
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1
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1, 1, 1, 1, 10, 1, 1, 47, 47, 1, 1, 176, 558, 176, 1, 1, 597, 4442, 4442, 597, 1, 1, 1926, 29247, 65812, 29247, 1926, 1, 1, 6043, 173385, 747931, 747931, 173385, 6043, 1, 1, 18652, 965620, 7279396, 13712662, 7279396, 965620, 18652, 1, 1, 56993, 5173340
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:
{1, 2, 12, 96, 912, 10080, 128160, 1854720, 30240000, 550126080,...}
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FORMULA
| p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}]
-2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).
Functional form:
p(x,n)=(3*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi[x, 1 - n, 1/2]
- 2*(-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog[ -n, x]/x).
t(n,m)=Coefficients(p(x,n))
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EXAMPLE
| {1},
{1, 1},
{1, 10, 1},
{1, 47, 47, 1},
{1, 176, 558, 176, 1},
{1, 597, 4442, 4442, 597, 1},
{1, 1926, 29247, 65812, 29247, 1926, 1},
{1, 6043, 173385, 747931, 747931, 173385, 6043, 1},
{1, 18652, 965620, 7279396, 13712662, 7279396, 965620, 18652, 1},
{1, 56993, 5173340, 64213532, 205619174, 205619174, 64213532, 5173340, 56993, 1}
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MATHEMATICA
| Clear[p, x, n]; p[x_, n_] = (3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}]
- 2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]
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CROSSREFS
| Sequence in context: A168524 A157277 A157629 * A174109 A171692 A152971
Adjacent sequences: A154333 A154334 A154335 * A154337 A154338 A154339
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KEYWORD
| nonn,tabl
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 07 2009
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