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A319934 Coefficients of the columns generating polynomials of the JacobiTheta3 array A319574 multiplied by n!, triangle read by rows, T(n,k) for 0 <= k <= n.

%I A319934 #10 Oct 07 2018 04:44:43
%S A319934 1,0,2,0,-4,4,0,16,-24,8,0,-48,176,-96,16,0,288,-1120,1120,-320,32,0,
%T A319934 -1920,8896,-11520,5440,-960,64,0,11520,-77952,127232,-80640,22400,
%U A319934 -2688,128,0,-80640,738048,-1480192,1195264,-448000,82432,-7168,256
%N A319934 Coefficients of the columns generating polynomials of the JacobiTheta3 array A319574 multiplied by n!, triangle read by rows, T(n,k) for 0 <= k <= n.
%C A319934 The purpose of the multiplication with n! is to make the coefficients integral.
%e A319934 Triangle starts:
%e A319934 [0] 1
%e A319934 [1] 0,      2
%e A319934 [2] 0,     -4,      4
%e A319934 [3] 0,     16,    -24,        8
%e A319934 [4] 0,    -48,    176,      -96,      16
%e A319934 [5] 0,    288,  -1120,     1120,    -320,      32
%e A319934 [6] 0,  -1920,   8896,   -11520,    5440,    -960,    64
%e A319934 [7] 0,  11520, -77952,   127232,  -80640,   22400, -2688,   128
%e A319934 [8] 0, -80640, 738048, -1480192, 1195264, -448000, 82432, -7168,  256
%p A319934 A319934poly := proc(N, opt) local a, n;
%p A319934 if   N = 0 then a := n -> 0!*1
%p A319934 elif N = 1 then a := n -> 1!*2*n
%p A319934 elif N = 2 then a := n -> 2!*2*n*(n-1)
%p A319934 elif N = 3 then a := n -> 3!*(4/3)*n*(n-1)*(n-2)
%p A319934 elif N = 4 then a := n -> 4!*(2/3)*n*(n^3-6*n^2+11*n-3)
%p A319934 elif N = 5 then a := n -> 5!*(4/15)*n*(n-1)*(n^3-9*n^2+26*n-9)
%p A319934 elif N = 6 then a := n -> 6!*(4/45)*n*(n-2)*(n-1)*(n^3-12*n^2+47*n-15)
%p A319934 elif N = 7 then a := n -> 7!*(8/315)*n*(n-1)*(n-2)*(n-3)*(n^3-15*n^2+74*n-15) fi;
%p A319934 if   opt = 'val' then print(seq(a(n), n=0..19))
%p A319934 elif opt = 'coe' then print(seq(coeff(a(n), n, i), i=0..N));
%p A319934 elif opt = 'pol' then sort(expand(a(n)), n, ascending) fi end:
%p A319934 for N from 0 to 7 do A319934poly(N, 'coe') od;
%Y A319934 Cf. A319574.
%K A319934 sign,tabl
%O A319934 0,3
%A A319934 _Peter Luschny_, Oct 02 2018