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A167008 a(n) = Sum_{k=0..n} C(n,k)^k. 9
1, 2, 4, 14, 106, 1732, 66634, 5745700, 1058905642, 461715853196, 461918527950694, 989913403174541980, 5009399946447021173140, 60070720443204091719085184, 1548154498059133199618813305334, 92346622775540905956057053976278584 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..75

V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

FORMULA

Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Dec 12 2012

EXAMPLE

The triangle of coefficients C(n,k)^k, n>=k>=0, begins:

1;

1, 1;

1, 2, 1;

1, 3, 9, 1;

1, 4, 36, 64, 1;

1, 5, 100, 1000, 625, 1;

1, 6, 225, 8000, 50625, 7776, 1;

1, 7, 441, 42875, 1500625, 4084101, 117649, 1;

1, 8, 784, 175616, 24010000, 550731776, 481890304, 2097152, 1; ...

in which the row sums form this sequence.

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n, k]^k, {k, 0, n}], {n, 1, 20}]}]

(* Program for numerical value of the limit a(n)^(1/n^2) *) (1-r)^(-r/2)/.FindRoot[(1-r)^(2*r-1)==r^(2*r), {r, 1/2}, WorkingPrecision->100] (* Vaclav Kotesovec, Dec 12 2012 *)

PROG

(PARI) a(n)=sum(k=0, n, binomial(n, k)^k)

(Haskell)

a167008 = sum . a219206_row  -- Reinhard Zumkeller, Feb 27 2015

CROSSREFS

Cf. A184731, A014062, A000169, A167009, A167010, A219206, A220359.

Sequence in context: A219767 A000609 A245079 * A238638 A240973 A102449

Adjacent sequences:  A167005 A167006 A167007 * A167009 A167010 A167011

KEYWORD

nonn,nice

AUTHOR

Paul D. Hanna, Nov 17 2009

STATUS

approved

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Last modified October 13 23:38 EDT 2019. Contains 327983 sequences. (Running on oeis4.)