OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) = coefficients [x^k]( p(x, n) ), where p(x, n) = (1/2)*( (1+x)^n + 2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ).
T(n, n-k) = T(n, k).
T(n, 1) = A003462(n), n >= 1. - G. C. Greubel, Oct 26 2022
EXAMPLE
Triangle of coefficients begins as:
1;
1, 1;
1, 4, 1;
1, 13, 13, 1;
1, 40, 118, 40, 1;
1, 121, 846, 846, 121, 1;
1, 364, 5279, 11784, 5279, 364, 1;
1, 1093, 30339, 129879, 129879, 30339, 1093, 1;
1, 3280, 165820, 1242672, 2337542, 1242672, 165820, 3280, 1;
MATHEMATICA
p[n_, x_]:= (1/2)*((1+x)^n +2^n*(1-x)^(n+1)*LerchPhi[x, -n, 1/2]);
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
PROG
(Magma) // As a triangle
LerchPhi:= func< x, n, q | (&+[x^k/(k+q)^n: k in [0..100]]) >;
p:= func< n, x | ( (x+1)^n + 2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) )/2 >;
R<x>:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( p(n, x) )): n in [0..12]]; // G. C. Greubel, Oct 26 2022
(SageMath)
def LerchPhi(x, n, q): return sum( x^k/(k+q)^n for k in range(100))
def p(n, x): return (1/2)*((1+x)^n +2^n*(1-x)^(n+1)*LerchPhi(x, -n, 1/2))
flatten([[( p(n, x) ).series(x, n+1).list()[k] for k in range(n+1)] for n in (0..12)]) # G. C. Greubel, Oct 26 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Nov 07 2008
STATUS
approved