A168524 Triangle of coefficients of g.f. a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = -2, b = 2, c = 1.

1, 1, 1, 1, 10, 1, 1, 39, 39, 1, 1, 120, 350, 120, 1, 1, 341, 2266, 2266, 341, 1, 1, 950, 12895, 28340, 12895, 950, 1, 1, 2659, 69201, 290891, 290891, 69201, 2659, 1, 1, 7540, 360772, 2661644, 4987254, 2661644, 360772, 7540, 1, 1, 21681, 1851948, 22618188, 72033750, 72033750, 22618188, 1851948, 21681, 1

Offset: 0

Author

Roger L. Bagula, Nov 28 2009

Keywords

• nonn
• tabl

Examples

			Triangle of coefficients begins as:
  1;
  1,     1;
  1,    10,       1;
  1,    39,      39,        1;
  1,   120,     350,      120,        1;
  1,   341,    2266,     2266,      341,        1;
  1,   950,   12895,    28340,    12895,      950,        1;
  1,  2659,   69201,   290891,   290891,    69201,     2659,       1;
  1,  7540,  360772,  2661644,  4987254,  2661644,   360772,    7540,     1;
  1, 21681, 1851948, 22618188, 72033750, 72033750, 22618188, 1851948, 21681, 1;
		

Links

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

Crossrefs

Cf. A168523, A168525.

Cf. A142458, A142459.

Programs

Mathematica

T[n_, a_, b_, c_]:= CoefficientList[Series[a*(1+x)^n + b*(1-x)^(n+2)* PolyLog[-n-1, x]/x + 2^n*c*(1-x)^(n+1)*LerchPhi[x, -n, 1/2], {x,0,30}], x];
Table[T[n, -2, 2, 1], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 19 2022 *)

Sage

m=12
def LerchPhi(x,s,a): return sum( x^j/(j+a)^s for j in (0..3*m) )
def p(n,x,a,b,c): return a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2)
def T(n,k,a,b,c): return ( p(n,x,a,b,c) ).series(x, n+1).list()[k]
flatten([[T(n,k,-2,2,1) for k in (0..n)] for n in (0..m)]) # G. C. Greubel, Mar 19 2022

Formula

From G. C. Greubel, Mar 19 2022: (Start)

G.f.: a*(1+x)^n + b*(1-x)^(n+2)*polylog(-n-1, x)/x + 2^n*c*(1-x)^(n+1)*LerchPhi(x, -n, 1/2), with a = -2, b = 2, c = 1.

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

Extensions

Edited by G. C. Greubel, Mar 19 2022