A050984 de Bruijn's S(5,n) = Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n, k)^5.

1, 30, 5730, 1696800, 613591650, 248832363780, 108702332138400, 50030418256790400, 23933662070438513250, 11795304320307625903500, 5952113838155498195161980, 3061813957188788125283450400, 1600318610176809076206888362400, 847745162264320796366122559544000

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

Generally (de Bruijn, 1958), S(s,n) is asymptotic to (2*cos(Pi/(2*s)))^(2*n*s+s-1)*2^(2-s)*(Pi*n)^((1-s)/2)*s^(-1/2). - Vaclav Kotesovec, Jul 09 2013

Andrews (1988) on page 162 states "If, however, we resort to the theory of hypergeometric series, we find that, for example, S(5,n) = - 5F_4[-2n,-2n,-2n,-2n,-2n 1,1,1,1 ; 1]". - _Michael Somos, Jul 24 2013

Examples

			1 + 30*x + 5730*x^2 + 1696800*x^3 + 613591650*x^4 + ...
		

References

• G. E. Andrews "Application of SCRATCHPAD to problems in special functions and combinatorics" Trends in Computer Algebra, R. Janssen, ed., Springer Lecture Notes in Comp.Sci., No. 296, pp. 159-166 (1988)
• N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Binomial Sums

Crossrefs

Cf. A000984, A006480, A050983, A227357.

Programs

Mathematica

Sum[ (-1)^(k+n)Binomial[ 2n, k ]^5, {k, 0, 2n} ]
a[ n_] := If[ n < 0, 0, (-1)^n HypergeometricPFQ[-2 n {1, 1, 1, 1, 1}, {1, 1, 1, 1}, 1]] (* Michael Somos, Jul 24 2013 *)

PARI

a(n)=sum(k=0,2*n,(-1)^(k+n)*binomial(2*n,k)^5) \\ Charles R Greathouse IV, Dec 21 2011

Formula

E.g.f.: Sum(n>=0,I^n*x^n/n!^5) * Sum(n>=0,(-I)^n*x^n/n!^5) = Sum(n>=0,a(n)*x^(2*n)/n!^5) where I^2=-1. - Paul D. Hanna, Dec 21 2011

a(n) ~ (5+sqrt(5))^(5*n+2)/(sqrt(5)*Pi^2*n^2*2^(5*(n+1))). - Vaclav Kotesovec, Jul 09 2013

Recurrence: n^4*(2*n - 1)^2*(220*n^3 - 858*n^2 + 1119*n - 488)*a(n) = 5*(110000*n^9 - 759000*n^8 + 2252400*n^7 - 3766690*n^6 + 3908325*n^5 - 2609510*n^4 + 1122418*n^3 - 300699*n^2 + 45738*n - 3024)*a(n-1) - 5*(2*n - 3)^2*(5*n - 8)*(5*n - 7)*(5*n - 6)*(5*n - 4)*(220*n^3 - 198*n^2 + 63*n - 7)*a(n-2). - Vaclav Kotesovec, Sep 27 2016

For n >= 1, a(n) = 2 * Sum_{k = 0..2*n-1} (-1)^(n+k) * binomial(2*n, k)^4 * binomial(2*n-1, k) = (1/n) * Sum_{k = 0..2*n} (-1)^(n+k) * k * binomial(2*n, k)^5. - Peter Bala, Oct 31 2024