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A168518
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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.
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5
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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
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OFFSET
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0,5
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LINKS
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FORMULA
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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.
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)
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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)) )
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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