A168517 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 = -1, b = 1, and c = 1, read by rows.

1, 1, 1, 1, 7, 1, 1, 27, 27, 1, 1, 87, 260, 87, 1, 1, 263, 1828, 1828, 263, 1, 1, 779, 11131, 24746, 11131, 779, 1, 1, 2299, 62793, 267515, 267515, 62793, 2299, 1, 1, 6799, 338902, 2529377, 4753074, 2529377, 338902, 6799, 1, 1, 20175, 1780242, 21935526, 70068408, 70068408, 21935526, 1780242, 20175, 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 = -1, b = 1, and c = 1.

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 = -1, b = 1, and c = 1.

T(n, n-k) = T(n, k). (End)

Example

Triangle begins as:
  1;
  1,     1;
  1,     7,       1;
  1,    27,      27,        1;
  1,    87,     260,       87,        1;
  1,   263,    1828,     1828,      263,        1;
  1,   779,   11131,    24746,    11131,      779,        1;
  1,  2299,   62793,   267515,   267515,    62793,     2299,       1;
  1,  6799,  338902,  2529377,  4753074,  2529377,   338902,    6799,     1;
  1, 20175, 1780242, 21935526, 70068408, 70068408, 21935526, 1780242, 20175, 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, -1, 1, 1], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)

Prog

(Sage)
def A168517(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([[A168517(n, k, -1, 1, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

Crossrefs

Cf. A168518, A168549, A168551, A168552.

Sequence in context: A218695 A179837 A238743 * A142465 A248829 A154337

Adjacent sequences: A168514 A168515 A168516 * A168518 A168519 A168520

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 28 2009

Extensions

Edited by G. C. Greubel, Mar 31 2022

Status

approved