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

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A142460, A155491, A155495, A157273, A166343.

Cf. A168517, A168549, A168551, A168552.

Sequence in context: A155495 A157273 A350729 * A174667 A174672 A174151

Adjacent sequences: A168515 A168516 A168517 * A168519 A168520 A168521

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 28 2009

Extensions

Edited by G. C. Greubel, Mar 31 2022

Status

approved