A326094 E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n!.

1, 5, 27, 185, 1693, 20565, 316375, 5948465, 133579065, 3517749125, 107024710675, 3714813650025, 145570443534805, 6383184292589525, 310815510350462415, 16694390352153656225, 983323269272332915825, 63186890982241624232325, 4409134435821084657726475, 332714992062735780407411225

Offset: 0

Author

Paul D. Hanna, Jun 21 2019

Keywords

• nonn

Comments

More generally, the following sums are equal:

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

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

here, q = (1+x) and p = 4, r = x.

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)^p * 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 = 1+x, p = 4, r = x, m = 1.

Examples

			E.g.f.: A(x) = 1 + 5*x + 27*x^2/2! + 185*x^3/3! + 1693*x^4/4! + 20565*x^5/5! + 316375*x^6/6! + 5948465*x^7/7! + 133579065*x^8/8! + 3517749125*x^9/9! + 107024710675*x^10/10! + ...
such that
A(x) = 1 + ((1+x) + 4)*x + ((1+x)^2 + 4)^2*x^2/2! + ((1+x)^3 + 4)^3*x^3/3! + ((1+x)^4 + 4)^4*x^4/4! + ((1+x)^5 + 4)^5*x^5/5! + ((1+x)^6 + 4)^6*x^6/6! + ((1+x)^7 + 4)^7*x^7/7! + ...
also
A(x) = 1 + (1+x)*exp(4*x*(1+x))*x + (1+x)^4*exp(4*x*(1+x)^2)*x^2/2! + (1+x)^9*exp(4*x*(1+x)^3)*x^3/3! + (1+x)^16*exp(4*x*(1+x)^4)*x^4/4! + (1+x)^25*exp(4*x*(1+x)^5)*x^5/5! + (1+x)^36*exp(4*x*(1+x)^6)*x^6/6! + ...
		

Links

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

Crossrefs

Cf. A326096, A326092, A326093.

Cf. A326274.

Programs

PARI

/* E.g.f.: Sum_{n>=0} ((1+x)^n + 4)^n * x^n/n! */
{a(n) = my(A = sum(m=0,n, ((1+x)^m + 4 +x*O(x^n))^m * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

PARI

/* E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(4*x*(1+x)^n) * x^n/n! */
{a(n) = my(A = sum(m=0,n, (1+x +x*O(x^n))^(m^2) * exp(4*x*(1+x)^m +x*O(x^n)) * x^m/m! )); n!*polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

Formula

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

E.g.f.: Sum_{n>=0} (1+x)^(n^2) * exp(4*x*(1+x)^n) * x^n/n!.

a(n) = 0 (mod 5) for n > 4.