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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
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: A344563 A167312 A114122 * A219836 A004174 A348874
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Oct 02 2018
STATUS
approved

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)