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

Author

Roger L. Bagula, Nov 29 2009

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

Cf. A001263, A132787.

Cf. A168517, A168518, A168549, A169552.

Programs

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 *)

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

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)

Extensions

Edited by G. C. Greubel, Mar 31 2022