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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]]. 0

%I

%S 1,1,1,1,8,1,1,25,25,1,1,78,234,78,1,1,239,1688,1688,239,1,1,724,

%T 10551,23560,10551,724,1,1,2181,60667,259743,259743,60667,2181,1,1,

%U 6554,331624,2485318,4675054,2485318,331624,6554,1,1,19675,1756354,21708014

%N 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]].

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

%F 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)).

%e {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}

%t 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[%]

%K nonn

%O 0,5

%A _Roger L. Bagula_, Nov 05 2008

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Last modified October 20 05:51 EDT 2020. Contains 337897 sequences. (Running on oeis4.)