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A005261 a(n) = Sum_{k = 0..n} C(n,k)^5.
(Formerly M2156)
9
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

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.

Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes.

Eric Weisstein's World of Mathematics, Binomial sums.

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

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

Cf. A000079, A000984, A000172, A005260, A069865.

Sequence in context: A092408 A180764 A228654 * A104898 A218432 A071799

Adjacent sequences:  A005258 A005259 A005260 * A005262 A005263 A005264

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Matthew Conroy, Mar 16 2006

STATUS

approved

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Last modified November 21 11:54 EST 2018. Contains 317447 sequences. (Running on oeis4.)