A147295 Pascal triangle shifted MacMahon numbers: p(x,n)=If[n < 2, -(-2)^n*(x - 1)^(n + 1)*LerchPhi[x, -n, 1/2], 2*x*(x + 1)^(n - 2) - (-2)^n*(x - 1)^(n + 1)*LerchPhi[x, -n, 1/2]].

1, 1, 1, 1, 8, 1, 1, 25, 25, 1, 1, 78, 234, 78, 1, 1, 239, 1688, 1688, 239, 1, 1, 724, 10551, 23560, 10551, 724, 1, 1, 2181, 60667, 259743, 259743, 60667, 2181, 1, 1, 6554, 331624, 2485318, 4675054, 2485318, 331624, 6554, 1, 1, 19675, 1756354, 21708014

Offset

0,5

Comments

Row sums are: {1, 2, 10, 52, 392, 3856, 46112, 645184, 10322048, 185794816, 3715891712}.

Links

Table of n, a(n) for n=0..48.

Formula

p(x,n)=If[n < 2, -(-2)^n*(x - 1)^(n + 1)*LerchPhi[x, -n, 1/2], 2*x*(x + 1)^(n - 2) - (-2)^n*(x - 1)^(n + 1)*LerchPhi[x, -n, 1/2]]; t(n,m)=coefficients(p(x,n)).

Example

{1}, {1, 1}, {1, 8, 1}, {1, 25, 25, 1}, {1, 78, 234, 78, 1}, {1, 239, 1688, 1688, 239, 1}, {1, 724, 10551, 23560, 10551, 724, 1}, {1, 2181, 60667, 259743, 259743, 60667, 2181, 1}, {1, 6554, 331624, 2485318, 4675054, 2485318, 331624, 6554, 1}, {1, 19675, 1756354, 21708014, 69413364, 69413364, 21708014, 1756354, 19675, 1}, {1, 59040, 9116157, 178300960, 906923394, 1527092608, 906923394, 178300960, 9116157, 59040, 1}

Mathematica

Clear[p, x, n]; p[x_, n_] = If[n < 2, -(-2)^n*(x - 1)^(n + 1)*LerchPhi[x, -n, 1/2], 2*x*(x + 1)^(n - 2) - (-2)^n*(x - 1)^(n + 1)*LerchPhi[x, -n, 1/2]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]

Crossrefs

Sequence in context: A358628 A167034 A155452 * A174388 A220718 A176283

Adjacent sequences: A147292 A147293 A147294 * A147296 A147297 A147298

Keyword

nonn

Author

Roger L. Bagula, Nov 05 2008

Status

approved