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A154337
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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}] -(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2.
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2
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1, 1, 1, 1, 7, 1, 1, 29, 29, 1, 1, 101, 312, 101, 1, 1, 327, 2372, 2372, 327, 1, 1, 1023, 15219, 34114, 15219, 1023, 1, 1, 3145, 88839, 381775, 381775, 88839, 3145, 1, 1, 9577, 490114, 3683815, 6934426, 3683815, 490114, 9577, 1, 1, 29003, 2610590, 32334362, 103464764, 103464764, 32334362, 2610590, 29003, 1
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OFFSET
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0,5
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COMMENTS
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Row sums are: {1, 2, 9, 60, 516, 5400, 66600, 947520, 15301440, 276877440,...}
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LINKS
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FORMULA
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p(x,n)=(3*(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)/2.
Functional form:
p(x,n)=(3*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) - (-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog( -n, x)/x)/2.
t(n,m)=Coefficients(p(x,n))
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EXAMPLE
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{1},
{1, 1},
{1, 7, 1},
{1, 29, 29, 1},
{1, 101, 312, 101, 1},
{1, 327, 2372, 2372, 327, 1},
{1, 1023, 15219, 34114, 15219, 1023, 1},
{1, 3145, 88839, 381775, 381775, 88839, 3145, 1},
{1, 9577, 490114, 3683815, 6934426, 3683815, 490114, 9577, 1},
{1, 29003, 2610590, 32334362, 103464764, 103464764, 32334362, 2610590, 29003, 1}
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MATHEMATICA
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Clear[p, x, n]; p[x_, n_] = (3*(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)/2;
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
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KEYWORD
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AUTHOR
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STATUS
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approved
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