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%I #9 Feb 27 2021 22:09:09
%S 224,1672,1672,10528,23528,10528,60636,259688,259688,60636,331584,
%T 2485232,4674944,2485232,331584,1756304,21707888,69413168,69413168,
%U 21707888,1756304,9116096,178300784,906923072,1527092216,906923072
%N Coefficients of Pascal's triangle polynomial minus MacMahon polynomial A060187 with minus the first and last row terms and powers of x divided out: f(n)=3^n - 2*n - 1; q(x,n)=2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2]; p(x,n)=((q[x, n] - (x + 1)^n)/x - f[n] - f[n]*x^(n - 2))/x.
%C Row sums starting with n=4 are {224, 3344, 44584, 640648, 10308576, 185754720, 3715772120}. First elements in each row are {224, 1672, 1672, 10528, 60636, 331584, 1756304, 9116096}. Subtracting out the row terms gives the middle elements of the difference.
%F f(n) = 3^n - 2*n - 1;
%F q(x,n) = 2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2];
%F p(x,n) = ((q[x, n] - (x + 1)^n)/x - f[n] - f[n]*x^(n - 2))/x;
%F t(n,m) = Coefficients(p(x,n)) with n starting at 4.
%e Triangle starts
%e {224},
%e {1672, 1672},
%e {10528, 23528, 10528},
%e {60636, 259688, 259688, 60636},
%e {331584, 2485232, 4674944, 2485232, 331584},
%e {1756304, 21707888, 69413168, 69413168, 21707888, 1756304},
%e {9116096, 178300784, 906923072, 1527092216, 906923072, 178300784, 9116096}
%t q[x_, n_] = 2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2]; p[x_, n_] = ((q[x, n] - (x + 1)^n)/x - f[n] - f[n]*x^(n - 2))/x; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 4, 10}]; Flatten[%]
%Y Cf. A061981, A060187.
%K nonn,tabl
%O 2,1
%A _Roger L. Bagula_, Nov 01 2008
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