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Search: id:A154335
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| A154335 |
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A triangular sequence of coefficients of polynomials: p(x,n)=(2*(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). |
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| 1, 1, 1, 1, 8, 1, 1, 35, 35, 1, 1, 126, 394, 126, 1, 1, 417, 3062, 3062, 417, 1, 1, 1324, 19895, 44680, 19895, 1324, 1, 1, 4111, 117021, 503827, 503827, 117021, 4111, 1, 1, 12602, 648616, 4882342, 9193838, 4882342, 648616, 12602, 1, 1, 38333, 3464840
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row sums are:
{1, 2, 10, 72, 648, 6960, 87120, 1249920, 20280960, 367960320,...}
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FORMULA
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p(x,n)=(2*(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).
Functional form:
p(x,n)=(2*(-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).
t(n,m)=Coefficients(p(x,n))
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EXAMPLE
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{1},
{1, 1},
{1, 8, 1},
{1, 35, 35, 1},
{1, 126, 394, 126, 1},
{1, 417, 3062, 3062, 417, 1},
{1, 1324, 19895, 44680, 19895, 1324, 1},
{1, 4111, 117021, 503827, 503827, 117021, 4111, 1},
{1, 12602, 648616, 4882342, 9193838, 4882342, 648616, 12602, 1},
{1, 38333, 3464840, 42960752, 137516234, 137516234, 42960752, 3464840, 38333, 1}
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MATHEMATICA
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Clear[p, x, n]; p[x_, n_] = (2*(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
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Sequence in context: A157208 A141686 A157148 this_sequence A142467 A142175 A142597
Adjacent sequences: A154332 A154333 A154334 this_sequence A154336 A154337 A154338
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KEYWORD
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nonn,tabl
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 07 2009
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