A168525 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 = 65/2, b = -162/2, c = 135/2.

19, 19, 19, 19, 146, 19, 19, 759, 759, 19, 19, 3154, 10374, 3154, 19, 19, 11543, 89398, 89398, 11543, 19, 19, 39210, 615669, 1394444, 615669, 39210, 19, 19, 127303, 3747297, 16267301, 16267301, 3747297, 127303, 19, 19, 401858, 21201472, 160611806, 302914330, 160611806, 21201472, 401858, 19

Offset: 0

Author

Roger L. Bagula, Nov 28 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  19;
  19,     19;
  19,    146,       19;
  19,    759,      759,        19;
  19,   3154,    10374,      3154,        19;
  19,  11543,    89398,     89398,     11543,        19;
  19,  39210,   615669,   1394444,    615669,     39210,       19;
  19, 127303,  3747297,  16267301,  16267301,   3747297,   127303,     19;
  19, 401858, 21201472, 160611806, 302914330, 160611806, 21201472, 401858, 19;
		

Links

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

Crossrefs

Cf. A142458, A142459.

Cf. A168523, A168524.

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, 65/2, -162/2, 135/2], {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,65/2, -162/2, 135/2) 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 = 65/2, b = -162/2, c = 135/2.

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

Extensions

Edited by G. C. Greubel, Mar 19 2022