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)^(n+1)*Sum_{k >= 0} (2*k+1)^(n-1)*x^k )^2.
T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (2^(n-1)*(1-x)^(n+2)*LerchPhi(x, -n+1, 1/2))^2.
Sum_{k=0..n} T(n, k) = 0^n.
T(n, n-k) = T(n, k). - G. C. Greubel, Mar 09 2022
EXAMPLE
Irregular triangle begins as:
1;
1, -2, 1;
1, 0, -2, 0, 1;
1, 10, 15, -52, 15, 10, 1;
1, 44, 484, -44, -970, -44, 484, 44, 1;
1, 150, 5933, 22792, 466, -58684, 466, 22792, 5933, 150, 1;
MATHEMATICA
p[n_, x_]:= p[n, x]= If[n==0, 1, (2^(n-1)*(1-x)^(n+1)*LerchPhi[x, -n+1, 1/2])^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*n+2)*sum( (2*j+1)^(n-1)*x^j for j in (0..2*n+2) )^2
def T(n, k): return ( p(n, x) ).series(x, 2*n+2).list()[k]
flatten([[T(n, k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2022
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Roger L. Bagula, Sep 29 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 09 2022
STATUS
approved