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

0,2

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

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000984, A006480, A050983, A227357.

Sequence in context: A204702 A206647 A321427 * A169686 A184889 A358481

Adjacent sequences: A050981 A050982 A050983 * A050985 A050986 A050987

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved