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Search: id:A154334
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| A154334 |
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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}] + (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2. |
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| 1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 51, 148, 51, 1, 1, 147, 992, 992, 147, 1, 1, 421, 5867, 12982, 5867, 421, 1, 1, 1213, 32475, 137671, 137671, 32475, 1213, 1, 1, 3527, 173110, 1286761, 2415602, 1286761, 173110, 3527, 1, 1, 10343, 902090, 11081582, 35361824
(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, 7, 36, 252, 2280, 25560, 342720, 5342400, 94711680,...}
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FORMULA
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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)/2.
Functional form:
p(x,n)=((-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, 5, 1},
{1, 17, 17, 1},
{1, 51, 148, 51, 1},
{1, 147, 992, 992, 147, 1},
{1, 421, 5867, 12982, 5867, 421, 1},
{1, 1213, 32475, 137671, 137671, 32475, 1213, 1},
{1, 3527, 173110, 1286761, 2415602, 1286761, 173110, 3527, 1},
{1, 10343, 902090, 11081582, 35361824, 35361824, 11081582, 902090, 10343, 1}
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MATHEMATICA
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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)/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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Sequence in context: A157637 A157181 A029847 this_sequence A144397 A047909 A111577
Adjacent sequences: A154331 A154332 A154333 this_sequence A154335 A154336 A154337
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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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