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A154338
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A triangular sequence of coefficients of polynomials: p(x,n)=(-(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, 1, 1, 1, 2, 1, 1, -1, -1, 1, 1, -24, -98, -24, 1, 1, -123, -1078, -1078, -123, 1, 1, -482, -8161, -18716, -8161, -482, 1, 1, -1685, -52071, -228485, -228485, -52071, -1685, 1, 1, -5548, -302396, -2308820, -4362634, -2308820, -302396, -5548, 1, 1, -17647
(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, 4, 0, -144, -2400, -36000, -564480, -9596160, -178536960,...}
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FORMULA
| p(x,n)=(-(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)=(-(-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, 2, 1},
{1, -1, -1, 1},
{1, -24, -98, -24, 1},
{1, -123, -1078, -1078, -123, 1},
{1, -482, -8161, -18716, -8161, -482,1},
{1, -1685, -52071, -228485, -228485, -52071, -1685, 1},
{1, -5548, -302396, -2308820, -4362634, -2308820, -302396, -5548, 1},
{1, -17647, -1660660, -20797588, -66792586, -66792586, -20797588, -1660660, -17647, 1}
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MATHEMATICA
| Clear[p, x, n]; p[x_, n_] = (-(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}]
- (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: A119335 A155869 A176564 * A087436 A053255 A085856
Adjacent sequences: A154335 A154336 A154337 * A154339 A154340 A154341
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KEYWORD
| sign,tabl,tabl
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 07 2009
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