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A335857
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a(n) is the determinant of the n X n Hankel matrix A with A(i,j) = A000108(i+j+6) for 0<=i,j<=n-1.
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0
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1, 132, 4719, 81796, 884884, 6852768, 41314284, 204951252, 869562265, 3245256300, 10880587575, 33309352440, 94307358288, 249485071616, 621856804272, 1470540624696, 3318218562009, 7179339254516, 14955909351383, 30104651175324, 58733021049780, 111358254207200
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OFFSET
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0,2
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COMMENTS
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Starting with 132 = A000108(6), the sequence is the Hankel transform of the Catalan numbers shifted to remove the first 6 terms.
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LINKS
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Table of n, a(n) for n=0..21.
Jishe Feng, The explicit formula of Hankel determinant with Catalan elements, arXiv:2010.06586 [math.GM], 2020.
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FORMULA
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a(n) = 2^5/(5 * 7!* 9!)*(n + 1)*(n + 2)^2*(n + 3)^3 *(n + 4)^2*(n + 5)*(2*n + 3)*(2*n + 5)^2*(2*n + 7)^2*(2*n + 9).
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EXAMPLE
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a(1) = 132 because 132 is the determinant of the 1 X 1 matrix [132].
a(2) = 4719 because 4719 is the determinant of the matrix
[ 132 429 ]
[ 429 1430 ].
a(3) = 81796 because 81796 is the determinant of the matrix
[ 132, 429, 1430 ]
[ 429, 1430, 4862 ]
[ 1430, 4862, 16796 ].
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PROG
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(PARI)
H(seq)={my(n=(#seq+1)\2); matdet(matrix(n, n, i, j, seq[i+j-1]))}
a(n, k=6)={H(vector(2*n, i, my(t=i+k-1); binomial(2*t, t)/(t+1)))} \\ Andrew Howroyd, Nov 26 2020
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CROSSREFS
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Hankel transforms of the Catalan numbers shifted by k for k=0..5 are A000012, A000012, A000027, A000330, A006858, A091962.
Cf. A000108 (Catalan numbers).
Sequence in context: A264179 A228333 A035837 * A270411 A258493 A184893
Adjacent sequences: A335854 A335855 A335856 * A335858 A335859 A335860
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KEYWORD
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nonn,easy
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AUTHOR
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Feng Jishe, Oct 12 2020
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STATUS
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approved
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