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 = -1, b = 1, c = 1.
T(n, n-k) = T(n, k). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 8, 1;
1, 31, 31, 1;
1, 98, 290, 98, 1;
1, 289, 1974, 1974, 289, 1;
1, 836, 11719, 25944, 11719, 836, 1;
1, 2419, 64929, 275307, 275307, 64929, 2419, 1;
1, 7046, 346192, 2573466, 4831134, 2573466, 346192, 7046, 1;
1, 20677, 1804144, 22163080, 70723522, 70723522, 22163080, 1804144, 20677, 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, -1, 1, 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, -1, 1, 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