OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = 3, b = -3, and c = 1.
From G. C. Greubel, Mar 31 2022: (Start)
T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = 3, b = -3, and c = 1.
T(n, n-k) = T(n, k). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 3, 1;
1, 11, 11, 1;
1, 43, 140, 43, 1;
1, 159, 1244, 1244, 159, 1;
1, 551, 8779, 19954, 8779, 551, 1;
1, 1819, 54249, 236347, 236347, 54249, 1819, 1;
1, 5811, 309742, 2353021, 4440834, 2353021, 309742, 5811, 1;
1, 18167, 1684634, 21025310, 67447952, 67447952, 21025310, 1684634, 18167, 1;
MATHEMATICA
p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x, -n-1, 1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x, -n, 1/2]);
Table[CoefficientList[p[x, n, 3, -3, 1], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Mar 31 2022 *)
PROG
(Sage)
def A168552(n, k, a, b, c): return (1/2)*( a*binomial(n, k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1, k-j)*(2*j+1)^n) for j in (0..k)) )
flatten([[A168552(n, k, 3, -3, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 31 2022
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Nov 29 2009
EXTENSIONS
Edited by G. C. Greubel, Mar 31 2022
STATUS
approved