A319934 - OEIS

%I #10 Oct 07 2018 04:44:43

%S 1,0,2,0,-4,4,0,16,-24,8,0,-48,176,-96,16,0,288,-1120,1120,-320,32,0,

%T -1920,8896,-11520,5440,-960,64,0,11520,-77952,127232,-80640,22400,

%U -2688,128,0,-80640,738048,-1480192,1195264,-448000,82432,-7168,256

%N 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 The purpose of the multiplication with n! is to make the coefficients integral.

%e Triangle starts:

%e [0] 1

%e [1] 0,      2

%e [2] 0,     -4,      4

%e [3] 0,     16,    -24,        8

%e [4] 0,    -48,    176,      -96,      16

%e [5] 0,    288,  -1120,     1120,    -320,      32

%e [6] 0,  -1920,   8896,   -11520,    5440,    -960,    64

%e [7] 0,  11520, -77952,   127232,  -80640,   22400, -2688,   128

%e [8] 0, -80640, 738048, -1480192, 1195264, -448000, 82432, -7168,  256

%p A319934poly := proc(N, opt) local a, n;

%p if   N = 0 then a := n -> 0!*1

%p elif N = 1 then a := n -> 1!*2*n

%p elif N = 2 then a := n -> 2!*2*n*(n-1)

%p elif N = 3 then a := n -> 3!*(4/3)*n*(n-1)*(n-2)

%p elif N = 4 then a := n -> 4!*(2/3)*n*(n^3-6*n^2+11*n-3)

%p elif N = 5 then a := n -> 5!*(4/15)*n*(n-1)*(n^3-9*n^2+26*n-9)

%p elif N = 6 then a := n -> 6!*(4/45)*n*(n-2)*(n-1)*(n^3-12*n^2+47*n-15)

%p 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 if   opt = 'val' then print(seq(a(n), n=0..19))

%p elif opt = 'coe' then print(seq(coeff(a(n), n, i), i=0..N));

%p elif opt = 'pol' then sort(expand(a(n)), n, ascending) fi end:

%p for N from 0 to 7 do A319934poly(N, 'coe') od;

%Y Cf. A319574.

%K sign,tabl

%O 0,3

%A _Peter Luschny_, Oct 02 2018