A251657 a(n) = (2^n + 3)^n.

1, 5, 49, 1331, 130321, 52521875, 90458382169, 662062621900811, 20248745068443234721, 2548385124666493326171875, 1305282261160894865367626964649, 2701607566979638625212777041914285051, 22497539334127167666989016452232087989410801, 751859086636251929847496735809485838154930419921875

Offset: 0

Author

Paul D. Hanna, Jan 29 2015

Keywords

• nonn

Comments

This is a special case of the more general statement:

Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1, b=3.

Examples

			E.g.f.: A(x) = 4^0 + 5^1*x + 7^2*x^2/2! + 11^3*x^3/3! + 19^4*x^4/4! + 35^5*x^5/5! + 67^6*x^6/6! + 131^7*x^7/7! +...+ (2^n+3)^n*x^n/n! +...
such that
A(x) = exp(3*x) + 2*exp(3*2*x) + 2^4*exp(3*4*x)*x^2/2! + 2^9*exp(3*8*x)*x^3/3! + 2^16*exp(3*16*x)*x^4/4! +...+ 2^(n^2)*exp(3*2^n*x)*x^n/n! +...
		

Crossrefs

Cf. A055601, A136516, A165327, A155810.

Programs

Mathematica

Table[(2^n+3)^n,{n,0,20}] (* Harvey P. Dale, Mar 16 2016 *)

PARI

{a(n,q=2,m=1,b=3) =( m*q^n + b)^n}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

PARI

{a(n,q=2,m=1,b=3) = sum(k=0,n, binomial(n,k) * b^k * m^(n-k) * (q^n)^(n-k))}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

PARI

{a(n,q=2,m=1,b=3) = n!*polcoeff(sum(k=0, n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

PARI

{a(n,q=2,m=1,b=3) = polcoeff(sum(k=0, n, m^k * q^(k^2) * x^k / (1 - b*q^k*x  +x*O(x^n))^(k+1) ), n)}
for(n=0,15,print1(a(n,q=2,m=1,b=3),", "))

Formula

E.g.f.: Sum_{n>=0} 2^(n^2) * exp(3*2^n*x) * x^n/n! = Sum_{n>=0} (2^n + 3)^n * x^n/n!.

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

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

a(n) = Sum_{k=0..n} A155810(k)*3^k.