OFFSET
0,3
LINKS
G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
FORMULA
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+4)*( Sum_{j >= 0} j^(n+1)*x^j )^2/x^2.
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+4)*( PolyLog(-n-1, x)/x)^2.
T(n, n-k) = T(n, k). - G. C. Greubel, Mar 09 2022
EXAMPLE
Irregular triangle begins as:
1;
1, 2, 1;
1, 8, 18, 8, 1;
1, 22, 143, 244, 143, 22, 1;
1, 52, 808, 3484, 5710, 3484, 808, 52, 1;
1, 114, 3853, 35032, 125746, 188908, 125746, 35032, 3853, 114, 1;
MATHEMATICA
p[n_, x_]:= p[n, x]= (1/x^2)*(1-x)^(2*n+4)*Sum[k^(n+1)*x^k, {k, 0, Infinity}]^2;
Table[CoefficientList[p[n, x], x], {n, 0, 12}]//Flatten (* modified by G. C. Greubel, Mar 09 2022 *)
PROG
(Sage)
def p(n, x): return (1/x^2)*(1-x)^(2*n+4)*sum( j^(n+1)*x^j for j in (0..2*n+4) )^2
def T(n, k): return ( p(n, x) ).series(x, 2*n+1).list()[k]
flatten([[T(n, k) for k in (0..2*n)] for n in (0..12)])
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Sep 29 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 09 2022
STATUS
approved