A358091 Triangle read by rows. Coefficients of the polynomials P(n, x) = 2^(n-2)*(3*n-1)* hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x). T(n, k) = [x^k] P(n, x).

1, 5, -6, 16, -60, 48, 44, -288, 660, -440, 112, -1056, 4032, -7280, 4368, 272, -3360, 17952, -52224, 81600, -45696, 640, -9792, 67200, -267520, 656640, -930240, 496128, 1472, -26880, 225216, -1133440, 3740352, -8160768, 10767680, -5537664

Offset: 1

Author

Peter Luschny, Oct 28 2022

Keywords

• sign
• tabl

Examples

			[1]    1;
[2]    5,     -6;
[3]   16,    -60,     48;
[4]   44,   -288,    660,     -440;
[5]  112,  -1056,   4032,    -7280,    4368;
[6]  272,  -3360,  17952,   -52224,   81600,   -45696;
[7]  640,  -9792,  67200,  -267520,  656640,  -930240,   496128;
[8] 1472, -26880, 225216, -1133440, 3740352, -8160768, 10767680, -5537664;
		

Crossrefs

Cf. A062236, A000309.

Programs

SageMath

def P(n):
    h = 2^(n-2)*(3*n-1)*hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x)
    return h.series(x, n+1).polynomial(SR)
for n in range(1, 9): print(P(n).list())
# To evaluate the polynomials use:
def p(n, t): return Integer(P(n)(x=t).n())
# For example the next statements yield A062236 and A000309.
print([p(n, -1/2) for n in range(1, 21)])
print([(-1)^n*p(n + 1, 1) for n in range(0, 22)])

Formula

P(n, -1/2) = A062236(n).

(-1)^n*P(n + 1, 1) = A000309(n).