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 A005261 a(n) = Sum_{k = 0..n} C(n,k)^5. (Formerly M2156) 18
 1, 2, 34, 488, 9826, 206252, 4734304, 113245568, 2816649826, 72001228052, 1883210876284, 50168588906768, 1357245464138656, 37198352117916992, 1030920212982957184, 28847760730478655488, 814066783370083977826 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) is the constant term in the expansion of ((1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n. - Seiichi Manyama, Oct 27 2019 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309. Eric Weisstein's World of Mathematics, Binomial sums. Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes. FORMULA a(n) ~ 4*5^(-1/2)*Pi^-2*n^-2*2^(5*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002 Recurrence (M. A. Perlstadt, 1987): 32*(55*n^2 + 33*n + 6)*(n - 1)^4*a(n-2) - (19415*n^6 - 27181*n^5 + 7453*n^4 + 3289*n^3 - 956*n^2 - 276*n + 96)*a(n-1) - (1155*n^6 + 693*n^5 - 732*n^4 - 715*n^3 + 45*n^2 + 210*n + 56)*a(n) + (55*n^2 - 77*n + 28)*(n + 1)^4*a(n+1) = 0. [Vaclav Kotesovec, Apr 27 2012] For r a nonnegative integer, Sum_{k = r..n} C(k,r)^5*C(n,k)^5 = C(n,r)^5*a(n-r), where we take a(n) = 0 for n < 0. - Peter Bala, Jul 27 2016 Sum_{n>=0} a(n) * x^n / (n!)^5 = (Sum_{n>=0} x^n / (n!)^5)^2. - Ilya Gutkovskiy, Jul 17 2020 MAPLE a := n -> hypergeom([seq(-n, i=1..5)], [seq(1, i=1..4)], -1): seq(simplify(a(n)), n=0..16); # Peter Luschny, Jul 27 2016 MATHEMATICA RecurrenceTable[{32*(55n^2+33n+6)*(n-1)^4*a[n-2]-(19415n^6-27181n^5+7453n^4+3289n^3-956n^2-276n+96)*a[n-1]-(1155n^6+693n^5-732n^4-715n^3+45n^2+210n+56)*a[n]+(55n^2-77n+28)*(n+1)^4*a[n+1]==0, a[0]==1, a[1]==2, a[2]==34}, a, {n, 0, 25}] (* or directly *) Table[Sum[Binomial[n, k]^5, {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 27 2012 *) PROG (PARI) a(n) = sum(k=0, n, binomial(n, k)^5); \\ Michel Marcus, Mar 09 2016 CROSSREFS Column k=5 of A309010. Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295. Sequence in context: A092408 A180764 A228654 * A104898 A218432 A071799 Adjacent sequences:  A005258 A005259 A005260 * A005262 A005263 A005264 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Matthew Conroy, Mar 16 2006 STATUS approved

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Last modified April 10 18:49 EDT 2021. Contains 342853 sequences. (Running on oeis4.)