A167008 a(n) = Sum_{k=0..n} C(n,k)^k.

1, 2, 4, 14, 106, 1732, 66634, 5745700, 1058905642, 461715853196, 461918527950694, 989913403174541980, 5009399946447021173140, 60070720443204091719085184, 1548154498059133199618813305334, 92346622775540905956057053976278584

Offset: 0

Author

Paul D. Hanna, Nov 17 2009

Keywords

• nonn
• nice

Comments

Row sums of A219206.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..75
• Vaclav Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013.

Crossrefs

Cf. A000169, A014062, A167009, A167010, A184731, A219206, A220359, A362288.

Programs

Haskell

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

Magma

[(&+[Binomial(n,j)^j: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Mathematica

Flatten[{1,Table[Sum[Binomial[n, k]^k, {k,0,n}], {n,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 *)
Total/@Table[Binomial[n,k]^k,{n,0,20},{k,0,n}] (* Harvey P. Dale, Oct 19 2021 *)

PARI

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

SageMath

[sum(binomial(n,j)^j for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Formula

Limit_{n->oo} a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Dec 12 2012