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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 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: A204702 A206647 A321427 * A169686 A184889 A358481 Adjacent sequences: A050981 A050982 A050983 * A050985 A050986 A050987 KEYWORD nonn AUTHOR Eric W. Weisstein STATUS approved

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