A326288 E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n!.

1, 4, 68, 2116, 98436, 6217924, 503491204, 50282169284, 6023071906180, 847321700204740, 137695169475601540, 25505309294030757316, 5326002105122774427524, 1242268006104279981404868, 321107726934189274515747460, 91359880704866957348006879172, 28441686041231472428045000672644, 9637951929231839144943126955386052, 3538621024404268912313596289954242692, 1401869934089183216934147248975602680260

Offset

0,2

Comments

More generally, the following sums are equal:

(1) Sum_{n>=0} (p + q^n)^n * r^n/n!,

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = exp(x) with p = -1, r = 4.

In general, let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =

Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);

here, F(x) = exp(x), q = exp(x), p = -1, r = 4, m = 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n!.

E.g.f.: Sum_{n>=0} 4^n * exp(n^2*x) * exp( -4*exp(n*x) ) / n!.

O.g.f.: Sum_{n>=0} 4^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).

a(n) = Sum_{k=0..n} 4^k * k^n * Stirling2(n,k).

Example

E.g.f.: A(x) = 1 + 4*x + 68*x^2/2! + 2116*x^3/3! + 98436*x^4/4! + 6217924*x^5/5! + 503491204*x^6/6! + 50282169284*x^7/7! + 6023071906180*x^8/8! + 847321700204740*x^9/9! + ...
such that
A(x) = 1 + 4*(exp(x) - 1) + 4^2*(exp(2*x) - 1)^2/2! + 4^3*(exp(3*x) - 1)^3/3! + 4^4*(exp(4*x) - 1)^4/4! + 4^5*(exp(5*x) - 1)^5/5! + 4^6*(exp(6*x) - 1)^6/6! + ...
also
A(x) = exp(-4) + 4*exp(x)*exp(-4*exp(x)) + 4^2*exp(4*x)*exp(-4*exp(2*x))/2! + 4^3*exp(9*x)*exp(-4*exp(3*x))/3! + 4^4*exp(16*x)*exp(-4*exp(4*x))/4! + 4^5*exp(25*x)*exp(-4*exp(5*x))/5! + 4^6*exp(36*x)*exp(-4*exp(6*x))/6! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 4*x + 68*x^2 + 2116*x^3 + 98436*x^4 + 6217924*x^5 + 503491204*x^6 + 50282169284*x^7 + 6023071906180*x^8 + 847321700204740*x^9 + ...
such that
B(x) = 1 + 4*x/(1-x) + 4^2*2^2*x^2/((1-2*x)*(1-4*x)) + 4^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 4^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 4^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...

Prog

(PARI) {a(n) = sum(k=0, n, 4^k * k^n * stirling(n, k, 2) )}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=0} 4^n * (exp(n*x) - 1)^n / n! */
{a(n) = n! * polcoeff(sum(m=0, n, 4^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* O.g.f.: Sum_{n>=0} 4^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
{a(n) = polcoeff(sum(m=0, n, 4^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A108459, A326270, A326271.

Sequence in context: A000658 A351027 A156470 * A302115 A009473 A009520

Adjacent sequences: A326285 A326286 A326287 * A326289 A326290 A326291

Keyword

nonn

Author

Paul D. Hanna, Jun 28 2019

Status

approved