A136507 a(n) = Sum_{k=0..n} binomial(2^(n-k) + k, n-k).

1, 3, 10, 71, 1925, 203904, 75214965, 94608676477, 409763735870986, 6208539881584781823, 334272186911271376874561, 64832512634295914941490910360, 45811927207957062190019240099653265

Offset: 0

Author

Paul D. Hanna, Jan 01 2008

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..59

Crossrefs

Cf. A014070 (C(2^n, n)), A136505 (C(2^n+1, n)), A136506 (C(2^n+2, n)).

Cf. A136508, A136509, A136555.

Programs

Magma

[(&+[Binomial(2^k +n-k, k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Mar 14 2021

Maple

A136507:= n-> add(binomial(2^k +n-k, k), k=0..n); seq(A136507(n), n=0..20); # G. C. Greubel, Mar 14 2021

Mathematica

Table[Sum[Binomial[2^(n-k)+k,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 08 2015 *)

PARI

{a(n)=sum(k=0,n,binomial(2^(n-k)+k,n-k))}
for(n=0,16, print1(a(n),", "))

PARI

/* a(n) = coefficient of x^n in o.g.f. series: */
{a(n)=polcoeff(sum(i=0,n,1/(1-x-2^i*x^2 +x*O(x^n))*log(1+2^i*x +x*O(x^n))^i/i!),n)}
for(n=0,16, print1(a(n),", "))

Sage

[sum(binomial(2^k +n-k, k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Mar 14 2021

Formula

G.f.: A(x) = Sum_{n>=0} (1 - x - 2^n*x^2)^(-1) * log(1 + 2^n*x)^n/n!.

a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

a(n) = Sum_{k=0..n} A136555(n-k+1, k). - G. C. Greubel, Mar 14 2021