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A319934
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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.
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1
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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
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OFFSET
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0,3
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COMMENTS
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The purpose of the multiplication with n! is to make the coefficients integral.
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LINKS
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EXAMPLE
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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
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MAPLE
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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;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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