A146745 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.

224, 1672, 1672, 10528, 23528, 10528, 60636, 259688, 259688, 60636, 331584, 2485232, 4674944, 2485232, 331584, 1756304, 21707888, 69413168, 69413168, 21707888, 1756304, 9116096, 178300784, 906923072, 1527092216, 906923072

Offset: 2

Author

Roger L. Bagula, Nov 01 2008

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle starts
{224},
{1672, 1672},
{10528, 23528, 10528},
{60636, 259688, 259688, 60636},
{331584, 2485232, 4674944, 2485232, 331584},
{1756304, 21707888, 69413168, 69413168, 21707888, 1756304},
{9116096, 178300784, 906923072, 1527092216, 906923072, 178300784, 9116096}
		

Crossrefs

Cf. A061981, A060187.

Programs

Mathematica

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[%]

Formula

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;

t(n,m) = Coefficients(p(x,n)) with n starting at 4.