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A099601 Quotient of de Bruijn sums S(4,n)/S(2,n). 7
1, 7, 131, 3067, 79459, 2181257, 62165039, 1818812387, 54257991011, 1642977121597, 50344383988381, 1557608560147757, 48577698917598031, 1525245771206644117, 48165918788138198759, 1528611371067309862067 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The de Bruijn sum S(s,n) = Sum_{k=0..2n} (-1)^(k+n) binomial(2n,k)^s.

Also the n-th term of the crystal ball sequence for the A_{2n} lattice.

REFERENCES

G. E. Andrews, Application of SCRATCHPAD to Problems in Special Functions and Combinatorics, in Trends in Computer Algebra, Springer-Verlag, 1988, pp. 158-166 MR0935413 (89c:05010).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

Recurrence: n^2*(2*n-1)^2*(48*n^2 - 126*n + 83)*a(n) = (6528*n^6 - 30192*n^5 + 55040*n^4 - 50406*n^3 + 24465*n^2 - 5985*n + 585)*a(n-1) - (n-1)^2*(2*n-3)^2*(48*n^2 - 30*n + 5)*a(n-2). - Vaclav Kotesovec, Mar 07 2014

a(n) ~ (1+sqrt(2))^(4*n+3/2) / (2^(9/4)*Pi*n). - Vaclav Kotesovec, Mar 07 2014

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n+k,2*n-k)*binomial(2*k,k). - Ilya Gutkovskiy, Nov 24 2017

a(n) = hypergeom([-2*n, -n, 2*n+1], [1, 1], 1). - Peter Luschny, Feb 13 2018

From Peter Bala, Dec 21 2020: (Start)

a(n) = Sum_{k = 0..n} C(2*n,n-k)^2 * C(2*n+k,k). Cf. A005258.

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

a(n) = C(2*n,n)^2 * hypergeom([-n, -n, 2*n+1], [n+1, n+1], 1).

a(n) = (-1)^n* C(2*n,n) * hypergeom([-n, 2*n+1, 2*n+1], [1, n+1], 1). (End)

a(n) = [x^n]  1/(1 - x)*( Legendre_P(m*n,(1 + x)/(1 - x)) ) at m = 2. At m = 1 we get the Apéry numbers A005258. - Peter Bala, Dec 23 2020

EXAMPLE

A003215(1)=7=a(1), A008384(2)=131=a(2), A008388(3)=3067=a(3), ...

MAPLE

a := n -> hypergeom([-2*n, -n, 2*n+1], [1, 1], 1):

seq(simplify(a(n)), n=0..15); # Peter Luschny, Feb 13 2018

MATHEMATICA

Table[Sum[(-1)^k*Binomial[2*n, k]^4, {k, 0, 2*n}] / Sum[(-1)^k*Binomial[2*n, k]^2, {k, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 07 2014 *)

PROG

(PARI) a(n)=if(n<0, 0, sum(k=0, 2*n, (-1)^k*binomial(2*n, k)^4)/ sum(k=0, 2*n, (-1)^k*binomial(2*n, k)^2))

CROSSREFS

Cf. A000984, A003215, A005258, A008384, A008388, A008392, A008396, A050983.

Sequence in context: A158701 A201308 A170912 * A028420 A220257 A220321

Adjacent sequences:  A099598 A099599 A099600 * A099602 A099603 A099604

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Oct 24 2004

STATUS

approved

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Last modified September 24 07:42 EDT 2021. Contains 347623 sequences. (Running on oeis4.)