A251184 a(n) = Sum_{k=0..n} binomial(n,k) * (2^k + 3)^k.

1, 6, 60, 1494, 135960, 53187306, 90775495620, 662696938144254, 20254044105203565360, 2548567387213968842106066, 1305307745923414524904985640540, 2701621925224675918174411993192242534, 22497571753504120612543889146545487252194120, 751859379104473999264213367292907652977053019173306

Offset

0,2

Links

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

Formula

G.f.: Sum_{n>=0} (2^n + 3)^n * x^n / (1-x)^(n+1).

G.f.: Sum_{n>=0} 2^(n^2) * x^n / (1-x - 3*2^n*x)^(n+1).

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

Example

G.f.: A(x) = 1 + 6*x + 60*x^2 + 1494*x^3 + 135960*x^4 + 53187306*x^5 +...
where we have the identity:
(1) A(x) = 1/(1-x) + (2+3)*x/(1-x)^2 + (2^2+3)^2*x^2/(1-x)^3 + (2^3+3)^3*x^3/(1-x)^4 + (2^4+3)^4*x^4/(1-x)^5 + (2^5+3)^5*x^5/(1-x)^6 +...
(2) A(x) = 1/(1-4*x) + 2*x/(1-x - 3*2*x)^2 + 2^4*x^2/(1-x - 3*2^2*x)^3 + 2^9*x^3/(1-x - 3*2^3*x)^4 + 2^16*x^4/(1-x - 3*2^4*x)^5 + 2^25*x^5/(1-x - 3*2^5*x)^6 +...
Illustration of initial terms.
a(0) = 1;
a(1) = 1 + (2+3) = 6;
a(2) = 1 + 2*(2+3) + (2^2+3)^2 = 60;
a(3) = 1 + 3*(2+3) + 3*(2^2+3)^2 + (2^3+3)^3 = 1494;
a(4) = 1 + 4*(2+3) + 6*(2^2+3)^2 + 4*(2^3+3)^3 + (2^4+3)^4 = 135960;
a(5) = 1 + 5*(2+3) + 10*(2^2+3)^2 + 10*(2^3+3)^3 + 5*(2^4+3)^4 + (2^5+3)^5 = 53187306; ...

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n, k) * (2^k + 3)^k )}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=sum(m=0, n, (2^m + 3)^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 - 3*2^m*x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A251182, A251183.

Sequence in context: A353196 A285955 A001416 * A329319 A351165 A003267

Adjacent sequences: A251181 A251182 A251183 * A251185 A251186 A251187

Keyword

nonn

Author

Paul D. Hanna, Jan 20 2015

Status

approved