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A108628 n-th term of the crystal ball sequence for A_{n+1} lattice for n>=0. 3
1, 7, 55, 471, 4251, 39733, 380731, 3716695, 36808723, 368750757, 3728940249, 38003358693, 389866749975, 4022124746409, 41697566691555, 434124925278807, 4536783726146499, 47569453938399445, 500266519237489357 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equals the secondary diagonal of square array A108625, in which row n equals the crystal ball sequence for A_n lattice. Main diagonal of square array A108625 equals the Apery numbers (A005258).

LINKS

Table of n, a(n) for n=0..18.

FORMULA

a(n) = Sum_{k = 0..n+1} C(n+1, k)^2 * C(n+k, k-1).

a(n) = A108625(n+1, n).

exp( Sum_{n >= 1} a(n-1)*x^n/n ) =  1 + x + 4*x^2 + 22*x^3 + 144*x^4 + 1048*x^5 + 8189*x^6 + 67325*x^7 + 574999*x^8 + ... appears to have integer coefficients. Cf. A208675. - Peter Bala, Jan 12 2016

Recurrence: (n+1)^2*(5*n^2 - 6*n + 2)*a(n) = (55*n^4 - 11*n^3 - 26*n^2 + 5*n + 5)*a(n-1) + (n-1)^2*(5*n^2 + 4*n + 1)*a(n-2). - Vaclav Kotesovec, Jan 13 2016

a(n) ~ sqrt(89/8 + 199/(8*sqrt(5))) * ((1+sqrt(5))/2)^(5*n) / (Pi*n). - Vaclav Kotesovec, Jan 13 2016

MATHEMATICA

Table[Sum[Binomial[n+1, k]^2 Binomial[n+k, k-1], {k, 0, n+1}], {n, 0, 20}] (* Harvey P. Dale, Apr 01 2013 *)

PROG

(PARI) a(n)=sum(k=0, n+1, binomial(n+1, k)^2*binomial(n+k, k-1))

CROSSREFS

Cf. A108625, A005258, A208675.

Sequence in context: A246459 A152262 A078018 * A116862 A096307 A199564

Adjacent sequences:  A108625 A108626 A108627 * A108629 A108630 A108631

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Jun 14 2005

STATUS

approved

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)