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A050984 de Bruijn's S(5,n). 7
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
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
Sequence in context: A204702 A206647 A321427 * A169686 A184889 A358481
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified July 24 21:32 EDT 2024. Contains 374585 sequences. (Running on oeis4.)