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A141660
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A generalization of the Eulerian numbers A123125 ( triangular sequence) to m: m=2; f(x, n) = (1 - m*x)^(n + 1)*Sum[k^n*(m*x)^k, {k, 0, Infinity}].
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0
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1, 0, 2, 0, 2, 4, 0, 2, 16, 8, 0, 2, 44, 88, 16, 0, 2, 104, 528, 416, 32, 0, 2, 228, 2416, 4832, 1824, 64, 0, 2, 480, 9528, 38656, 38112, 7680, 128, 0, 2, 988, 34344, 249904, 499808, 274752, 31616, 256, 0, 2, 2008, 116864, 1411744, 4998080, 5646976, 1869824
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OFFSET
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1,3
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COMMENTS
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Row sums are:
{1, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126}.
This sequence is just a substitution of : x->m*x.
Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,6,...] DELTA [2,0,4,0,6,0,8,0,10,0,...] where DELTA is the operator defined in A084938 . [From Philippe DELEHAM, Sep 15 2008]
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LINKS
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Table of n, a(n) for n=1..53.
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FORMULA
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m=2; f(x, n) = (1 - m*x)^(n + 1)*Sum[k^n*(m*x)^k, {k, 0, Infinity}]; t(n,m)=Coefficients(f(x,n)).
T(n,k)=2^k*A123125(n,k). [From Philippe DELEHAM, Sep 15 2008]
Sum_{k, 0<=k<=n} T(n,k) = A000629(n).- From Philippe Deléham, Nov 19 2011.
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EXAMPLE
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{1},
{0, 2},
{0, 2, 4},
{0, 2, 16, 8},
{0, 2, 44, 88, 16},
{0, 2, 104, 528, 416, 32},
{0, 2, 228, 2416, 4832, 1824, 64},
{0, 2, 480, 9528, 38656, 38112, 7680, 128},
{0, 2, 988, 34344, 249904, 499808, 274752, 31616, 256},
{0, 2, 2008, 116864, 1411744, 4998080, 5646976, 1869824, 128512, 512},
{0, 2, 4052, 382720, 7283072, 41931328, 83862656, 58264576, 12247040, 518656, 1024}
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MATHEMATICA
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Clear[f, x, n, a, m] m = 2; f[x_, n_] := f[x, n] = (1 - m*x)^(n + 1)*Sum[k^n*(m*x)^k, {k, 0, Infinity}] Table[FullSimplify[ExpandAll[f[x, n]]], {n, 0, 10}]; a = Table[CoefficientList[FullSimplify[ExpandAll[f[x, n]]], x], {n, 0, 10}]; Flatten[a]
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CROSSREFS
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Cf. A123125.
Sequence in context: A209691 A221683 A199335 * A209699 A115273 A194759
Adjacent sequences: A141657 A141658 A141659 * A141661 A141662 A141663
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula and Gary W. Adamson, Sep 04 2008
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EXTENSIONS
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Corrected example . - Philippe DELEHAM, Oct 16 2008
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STATUS
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approved
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