%I #7 Mar 20 2022 02:19:07

%S 1,1,1,1,10,1,1,39,39,1,1,120,350,120,1,1,341,2266,2266,341,1,1,950,

%T 12895,28340,12895,950,1,1,2659,69201,290891,290891,69201,2659,1,1,

%U 7540,360772,2661644,4987254,2661644,360772,7540,1,1,21681,1851948,22618188,72033750,72033750,22618188,1851948,21681,1

%N 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.

%H G. C. Greubel, <a href="/A168524/b168524.txt">Rows n = 0..50 of the triangle, flattened</a>

%F From _G. C. Greubel_, Mar 19 2022: (Start)

%F 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.

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

%e Triangle of coefficients begins as:

%e 1;

%e 1, 1;

%e 1, 10, 1;

%e 1, 39, 39, 1;

%e 1, 120, 350, 120, 1;

%e 1, 341, 2266, 2266, 341, 1;

%e 1, 950, 12895, 28340, 12895, 950, 1;

%e 1, 2659, 69201, 290891, 290891, 69201, 2659, 1;

%e 1, 7540, 360772, 2661644, 4987254, 2661644, 360772, 7540, 1;

%e 1, 21681, 1851948, 22618188, 72033750, 72033750, 22618188, 1851948, 21681, 1;

%t 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];

%t Table[T[n, -2, 2, 1], {n, 0, 12}]//Flatten (* modified by _G. C. Greubel_, Mar 19 2022 *)

%o (Sage)

%o m=12

%o def LerchPhi(x,s,a): return sum( x^j/(j+a)^s for j in (0..3*m) )

%o 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)

%o def T(n,k,a,b,c): return ( p(n,x,a,b,c) ).series(x, n+1).list()[k]

%o flatten([[T(n,k,-2,2,1) for k in (0..n)] for n in (0..m)]) # _G. C. Greubel_, Mar 19 2022

%Y Cf. A168523, A168525.

%Y Cf. A142458, A142459.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Nov 28 2009

%E Edited by _G. C. Greubel_, Mar 19 2022