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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).

%I A358091 #9 Oct 28 2022 10:10:29
%S A358091 1,5,-6,16,-60,48,44,-288,660,-440,112,-1056,4032,-7280,4368,272,
%T A358091 -3360,17952,-52224,81600,-45696,640,-9792,67200,-267520,656640,
%U A358091 -930240,496128,1472,-26880,225216,-1133440,3740352,-8160768,10767680,-5537664
%N 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).
%F A358091 P(n, -1/2) = A062236(n).
%F A358091 (-1)^n*P(n + 1, 1) = A000309(n).
%e A358091 [1]    1;
%e A358091 [2]    5,     -6;
%e A358091 [3]   16,    -60,     48;
%e A358091 [4]   44,   -288,    660,     -440;
%e A358091 [5]  112,  -1056,   4032,    -7280,    4368;
%e A358091 [6]  272,  -3360,  17952,   -52224,   81600,   -45696;
%e A358091 [7]  640,  -9792,  67200,  -267520,  656640,  -930240,   496128;
%e A358091 [8] 1472, -26880, 225216, -1133440, 3740352, -8160768, 10767680, -5537664;
%o A358091 (SageMath)
%o A358091 def P(n):
%o A358091     h = 2^(n-2)*(3*n-1)*hypergeometric([-3*n, 1 - n, -n + 4/3], [-n, -n + 1/3], x)
%o A358091     return h.series(x, n+1).polynomial(SR)
%o A358091 for n in range(1, 9): print(P(n).list())
%o A358091 # To evaluate the polynomials use:
%o A358091 def p(n, t): return Integer(P(n)(x=t).n())
%o A358091 # For example the next statements yield A062236 and A000309.
%o A358091 print([p(n, -1/2) for n in range(1, 21)])
%o A358091 print([(-1)^n*p(n + 1, 1) for n in range(0, 22)])
%Y A358091 Cf. A062236, A000309.
%K A358091 sign,tabl
%O A358091 1,2
%A A358091 _Peter Luschny_, Oct 28 2022