A168518 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2, read by rows.
1, 1, 1, 1, 12, 1, 1, 51, 51, 1, 1, 170, 514, 170, 1, 1, 521, 3646, 3646, 521, 1, 1, 1552, 22247, 49472, 22247, 1552, 1, 1, 4591, 125565, 534995, 534995, 125565, 4591, 1, 1, 13590, 677776, 5058698, 9506078, 5058698, 677776, 13590, 1, 1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 12, 1; 1, 51, 51, 1; 1, 170, 514, 170, 1; 1, 521, 3646, 3646, 521, 1; 1, 1552, 22247, 49472, 22247, 1552, 1; 1, 4591, 125565, 534995, 534995, 125565, 4591, 1; 1, 13590, 677776, 5058698, 9506078, 5058698, 677776, 13590, 1; 1, 40341, 3560448, 43870968, 140136690, 140136690, 43870968, 3560448, 40341, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Mathematica
p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]); Table[CoefficientList[p[x,n,-4,2,2], x], {n,0,10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *) -
Sage
def A168518(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ) flatten([[A168518(n,k,-4,2,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022
Formula
G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -4, b = 2, and c = 2.
From G. C. Greubel, Mar 31 2022: (Start)
T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = -4, b = 2, and c = 2.
T(n, n-k) = T(n, k). (End)
Extensions
Edited by G. C. Greubel, Mar 31 2022