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A243918 a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 2^k)^k. 6
1, 4, 32, 814, 86600, 39560554, 75654970772, 594996059517934, 19035905851947436400, 2460857798358946973785234, 1280109151917797032199865564812, 2672783800502564772495577135824089014, 22366199286781599568269093307412768076442280 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..12.

FORMULA

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

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

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

E.g.f.: exp(x) * Sum_{n>=0} (1 + 2^n)^n * x^n / n!.

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jun 18 2014

EXAMPLE

O.g.f.: A(x) = 1 + 4*x + 32*x^2 + 814*x^3 + 86600*x^4 + 39560554*x^5 +...

where the g.f. may be expressed by the series identity:

A(x) = 1/(1-x) + 3*x/(1-x)^2 + 5^2*x^2/(1-x)^3 + 9^3*x^3/(1-x)^4 + 17^4*x^4/(1-x)^5 + 33^5*x^5/(1-x)^6 + 65^6*x^6/(1-x)^7 +...

A(x) = 1/(1-2*x) + 2*x/(1-3*x)^2 + 2^4*x^2/(1-5*x)^3 + 2^9*x^3/(1-9*x)^4 + 2^16*x^4/(1-17*x)^5 + 2^25*x^5/(1-33*x)^6 + 2^36*x^6/(1-65*x)^7 +...

Illustration of initial terms:

a(0) = 1;

a(1) = 1 + (1+2);

a(2) = 1 + 2*(1+2) + (1+2^2)^2;

a(3) = 1 + 3*(1+2) + 3*(1+2^2)^2 + (1+2^3)^3;

a(4) = 1 + 4*(1+2) + 6*(1+2^2)^2 + 4*(1+2^3)^3 + (1+2^4)^4;

a(5) = 1 + 5*(1+2) + 10*(1+2^2)^2 + 10*(1+2^3)^3 + 5*(1+2^4)^4 + (1+2^5)^5; ...

Also, by a binomial identity we have

a(0) = 1;

a(1) = 2 + 2;

a(2) = 2^2 + 2*(1+2)*2 + 2^4;

a(3) = 2^3 + 3*(1+2)^2*2 + 3*(1+2^2)*2^4 + 2^9;

a(4) = 2^4 + 4*(1+2)^3*2 + 6*(1+2^2)^2*2^4 + 4*(1+2^3)*2^9 + 2^16;

a(5) = 2^5 + 5*(1+2)^4*2 + 10*(1+2^2)^3*2^4 + 10*(1+2^3)^2*2^9 + 5*(1+2^4)*2^16 + 2^25; ...

MATHEMATICA

Table[Sum[Binomial[n, k]*(1+2^k)^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 18 2014 *)

PROG

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(1+2^k)^k)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(1+2^k)^(n-k)*2^(k^2))}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A244004, A136516.

Sequence in context: A134087 A132854 A136471 * A298896 A231991 A028369

Adjacent sequences:  A243915 A243916 A243917 * A243919 A243920 A243921

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 17 2014

STATUS

approved

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Last modified February 22 09:17 EST 2020. Contains 332133 sequences. (Running on oeis4.)