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A335857 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. 0
1, 132, 4719, 81796, 884884, 6852768, 41314284, 204951252, 869562265, 3245256300, 10880587575, 33309352440, 94307358288, 249485071616, 621856804272, 1470540624696, 3318218562009, 7179339254516, 14955909351383, 30104651175324, 58733021049780, 111358254207200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Starting with 132 = A000108(6), the sequence is the Hankel transform of the Catalan numbers shifted to remove the first 6 terms.

LINKS

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.

FORMULA

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).

EXAMPLE

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 ].

PROG

(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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Feng Jishe, Oct 12 2020

STATUS

approved

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Last modified March 5 14:57 EST 2021. Contains 341823 sequences. (Running on oeis4.)