A168551 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, 5, 1, 1, 19, 19, 1, 1, 65, 200, 65, 1, 1, 211, 1536, 1536, 211, 1, 1, 665, 9955, 22350, 9955, 665, 1, 1, 2059, 58521, 251931, 251931, 58521, 2059, 1, 1, 6305, 324322, 2441199, 4596954, 2441199, 324322, 6305, 1, 1, 19171, 1732438, 21480418, 68758180, 68758180, 21480418, 1732438, 19171, 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,     5,       1;
  1,    19,      19,        1;
  1,    65,     200,       65,        1;
  1,   211,    1536,     1536,      211,        1;
  1,   665,    9955,    22350,     9955,      665,        1;
  1,  2059,   58521,   251931,   251931,    58521,     2059,       1;
  1,  6305,  324322,  2441199,  4596954,  2441199,   324322,    6305,     1;
  1, 19171, 1732438, 21480418, 68758180, 68758180, 21480418, 1732438, 19171, 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 A168552(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([[A168552(n, k, 1, -1, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022

Crossrefs

Cf. A001263, A132787.

Cf. A168517, A168518, A168549, A169552.

Sequence in context: A029847 A154334 A178346 * A262307 A144397 A047909

Adjacent sequences: A168548 A168549 A168550 * A168552 A168553 A168554

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 29 2009

Extensions

Edited by G. C. Greubel, Mar 31 2022

Status

approved