OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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)
EXAMPLE
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;
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Nov 28 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 19 2022
STATUS
approved