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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. 1
1, 0, 2, 0, -4, 4, 0, 16, -24, 8, 0, -48, 176, -96, 16, 0, 288, -1120, 1120, -320, 32, 0, -1920, 8896, -11520, 5440, -960, 64, 0, 11520, -77952, 127232, -80640, 22400, -2688, 128, 0, -80640, 738048, -1480192, 1195264, -448000, 82432, -7168, 256 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..44.

EXAMPLE

Triangle starts:

[0] 1

[1] 0,      2

[2] 0,     -4,      4

[3] 0,     16,    -24,        8

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

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

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

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

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

MAPLE

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

if   N = 0 then a := n -> 0!*1

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

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

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

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

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

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

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;

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

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

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

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

CROSSREFS

Cf. A319574.

Sequence in context: A103328 A167312 A114122 * A219836 A004174 A300328

Adjacent sequences:  A319931 A319932 A319933 * A319935 A319936 A319937

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Oct 02 2018

STATUS

approved

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Last modified March 29 15:03 EDT 2020. Contains 333107 sequences. (Running on oeis4.)