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A050984 de Bruijn's S(5,n). 6
1, 30, 5730, 1696800, 613591650, 248832363780, 108702332138400, 50030418256790400, 23933662070438513250, 11795304320307625903500, 5952113838155498195161980, 3061813957188788125283450400, 1600318610176809076206888362400, 847745162264320796366122559544000 (list; graph; refs; listen; history; text; internal format)
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

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: A204975 A204702 A206647 * A169686 A184889 A300147

Adjacent sequences:  A050981 A050982 A050983 * A050985 A050986 A050987

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified October 21 17:10 EDT 2018. Contains 316427 sequences. (Running on oeis4.)