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A302909
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Determinant of n X n matrix whose main diagonal consists of the first n 5-gonal numbers and all other elements are 1's.
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6
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1, 4, 44, 924, 31416, 1570800, 108385200, 9863053200, 1144114171200, 164752440652800, 28831677114240000, 6025820516876160000, 1482351847151535360000, 423952628285339112960000, 139480414705876568163840000, 52305155514703713061440000000
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Gamma(n) * Gamma(n + 5/3) * 3^(n + 1) / (5 * Gamma(2/3) * 2^n).
a(n) ~ Gamma(1/3) * 3^(n + 3/2) * n^(2*n + 2/3) / (5 * 2^n * exp(2*n)).
(End)
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EXAMPLE
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The 7 X 7 matrix (as below) has determinant 108385200.
1 1 1 1 1 1 1
1 5 1 1 1 1 1
1 1 12 1 1 1 1
1 1 1 22 1 1 1
1 1 1 1 35 1 1
1 1 1 1 1 51 1
1 1 1 1 1 1 70
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MAPLE
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d:=(i, j)->`if`(i<>j, 1, i*(3*i-1)/2):
seq(LinearAlgebra[Determinant](Matrix(n, d)), n=1..17);
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MATHEMATICA
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Table[FullSimplify[Gamma[n] * Gamma[n + 5/3] * 3^(n + 1) / (5 * Gamma[2/3] * 2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)
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PROG
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(PARI) a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-1)/2))); \\ Michel Marcus, Apr 16 2018
(PARI) first(n) = my(res = vector(n)); res[1] = 1; for(i = 1, n - 1, res[i + 1] = res[i] * i*(3*i + 5)/2); res \\ David A. Corneth, Apr 16 2018
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CROSSREFS
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Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), this sequence (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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