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 A141660 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}]. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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] LINKS FORMULA 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 DelĂ©ham, Nov 19 2011. EXAMPLE {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} MATHEMATICA 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] CROSSREFS Cf. A123125. Sequence in context: A209691 A221683 A199335 * A209699 A115273 A194759 Adjacent sequences:  A141657 A141658 A141659 * A141661 A141662 A141663 KEYWORD nonn,uned,tabl AUTHOR Roger L. Bagula and Gary W. Adamson, Sep 04 2008 EXTENSIONS Corrected example . - Philippe DELEHAM, Oct 16 2008 STATUS approved

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