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%I #9 Oct 28 2022 10:10:29
%S 1,5,-6,16,-60,48,44,-288,660,-440,112,-1056,4032,-7280,4368,272,
%T -3360,17952,-52224,81600,-45696,640,-9792,67200,-267520,656640,
%U -930240,496128,1472,-26880,225216,-1133440,3740352,-8160768,10767680,-5537664
%N 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).
%F P(n, -1/2) = A062236(n).
%F (-1)^n*P(n + 1, 1) = A000309(n).
%e [1] 1;
%e [2] 5, -6;
%e [3] 16, -60, 48;
%e [4] 44, -288, 660, -440;
%e [5] 112, -1056, 4032, -7280, 4368;
%e [6] 272, -3360, 17952, -52224, 81600, -45696;
%e [7] 640, -9792, 67200, -267520, 656640, -930240, 496128;
%e [8] 1472, -26880, 225216, -1133440, 3740352, -8160768, 10767680, -5537664;
%o (SageMath)
%o def P(n):
%o h = 2^(n-2)*(3*n-1)*hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x)
%o return h.series(x, n+1).polynomial(SR)
%o for n in range(1, 9): print(P(n).list())
%o # To evaluate the polynomials use:
%o def p(n, t): return Integer(P(n)(x=t).n())
%o # For example the next statements yield A062236 and A000309.
%o print([p(n, -1/2) for n in range(1, 21)])
%o print([(-1)^n*p(n + 1, 1) for n in range(0, 22)])
%Y Cf. A062236, A000309.
%K sign,tabl
%O 1,2
%A _Peter Luschny_, Oct 28 2022
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