A251182 a(n) = Sum_{k=0..n} binomial(n, k) * (2^k - 1)^k.

1, 2, 12, 374, 52056, 28885802, 62696043492, 533314127677214, 17882368106959072176, 2375841795610783881752402, 1255349217681407318492850586812, 2644225793295900889867998778302561734, 22235530372401312606563659670627119777739016, 747102526634678016053120249315815798707672485729402

Offset

0,2

Links

Table of n, a(n) for n=0..13.

Formula

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

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

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jan 25 2015

Example

G.f.: A(x) = 1 + 2*x + 12*x^2 + 374*x^3 + 52056*x^4 + 28885802*x^5 +...
where we have the identity:
(1) A(x) = 1/(1-x) + (2-1)*x/(1-x)^2 + (2^2-1)^2*x^2/(1-x)^3 + (2^3-1)^3*x^3/(1-x)^4 + (2^4-1)^4*x^4/(1-x)^5 + (2^5-1)^5*x^5/(1-x)^6 +...
(2) A(x) = 1 + 2*x/(1-x + 2*x)^2 + 2^4*x^2/(1-x + 2^2*x)^3 + 2^9*x^3/(1-x + 2^3*x)^4 + 2^16*x^4/(1-x + 2^4*x)^5 + 2^25*x^5/(1-x + 2^5*x)^6 + 2^36*x^6/(1-x + 2^6*x)^7 +...
Illustration of initial terms:
a(0) = 1;
a(1) = 1 + (2-1) = 2;
a(2) = 1 + 2*(2-1) + (2^2-1)^2 = 12;
a(3) = 1 + 3*(2-1) + 3*(2^2-1)^2 + (2^3-1)^3 = 374;
a(4) = 1 + 4*(2-1) + 6*(2^2-1)^2 + 4*(2^3-1)^3 + (2^4-1)^4 = 52056;
a(5) = 1 + 5*(2-1) + 10*(2^2-1)^2 + 10*(2^3-1)^3 + 5*(2^4-1)^4 + (2^5-1)^5 = 28885802; ...

Mathematica

Table[1 + Sum[Binomial[n, k] * (2^k-1)^k, {k, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jan 25 2015 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n, k) * (2^k - 1)^k )}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=sum(m=0, n, (2^m - 1)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=sum(m=0, n, 2^(m^2) * x^m / (1-x + x*2^m +x*O(x^n))^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A251183, A251184.

Sequence in context: A131690 A158261 A215956 * A266241 A012547 A009706

Adjacent sequences: A251179 A251180 A251181 * A251183 A251184 A251185

Keyword

nonn

Author

Paul D. Hanna, Jan 19 2015

Status

approved