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.

1, 132, 4719, 81796, 884884, 6852768, 41314284, 204951252, 869562265, 3245256300, 10880587575, 33309352440, 94307358288, 249485071616, 621856804272, 1470540624696, 3318218562009, 7179339254516, 14955909351383, 30104651175324, 58733021049780, 111358254207200

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

Jishe Feng, The explicit formula of Hankel determinant with Catalan elements, arXiv:2010.06586 [math.GM], 2020.

Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

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).

From G. C. Greubel, Dec 17 2021: (Start)

a(n) = binomial(n+5, 5)*binomial(2*n+7, 3)*binomial(2*n+9, 7)/1260.

G.f.: (1 + 116*x + 2727*x^2 + 21572*x^3 + 70328*x^4 + 103376*x^5 + 70328*x^6 + 21572*x^7 + 2727*x^8 + 116*x^9 + x^10)/(1 - x)^16. (End)

a(n) = -a(-6-n) for all n in Z. - Michael Somos, Jun 27 2023

a(n) ~ n^15/4465125. - Stefano Spezia, Dec 09 2023

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 ].
G.f. = 1 + 132*x + 4719*x^2 + 81796*x^3 + 884884*x^4 + ... - Michael Somos, Jun 27 2023

Mathematica

Table[Binomial[n+5, 5]*Binomial[2*n+7, 3]*Binomial[2*n+9, 7]/1260, {n, 0, 30}] (* G. C. Greubel, Dec 17 2021 *)
a[ n_] := Binomial[n+5, 4]*Binomial[2*n+7, 3]*Binomial[2*n+9, 8]/1575; (* Michael Somos, Jun 27 2023 *)

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
(PARI) {a(n) = prod(k=1, 5, (n+k)^min(6-k, k)) * prod(k=1, 4, (2*n+2*k+1)^min(5-k, k))/285768000}; /* Michael Somos, Jun 27 2023 */
(Sage) [binomial(n+5, 5)*binomial(2*n+7, 3)*binomial(2*n+9, 7)/1260 for n in (0..30)] # G. C. Greubel, Dec 17 2021

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