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
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
Programs
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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