A393754 E.g.f.: exp( x * Sum_{n>=0} n! * 4^n * x^n ).

1, 1, 9, 217, 10225, 801681, 94616761, 15696482089, 3483330195297, 996272567289505, 356793033395698921, 156361716462978988281, 82299934985279126998609, 51225831690387051722811697, 37215503820999440924493944985, 31207282397036870862892419982921, 29915379892820266366998272403668161

Offset

0,3

Links

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

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! along with F(x) = Sum_{n>=0} n! * x^n satisfy the following formulas.

(1) A(x) = exp( x*F(4*x) ).

(2) A(x) = exp(x + 4*x^2 * A'(x)/A(x)).

(3) A(x) = G(4*x)^(1/4) where G(x) is the g.f. of A158876.

(4) A'(x)/A(x) = Sum_{n>=0} (n+1)! * 4^n * x^n.

(5) [x^n] A(x)^(4*n) * (2 - F(4*x)) = 0 for n > 0.

(6) [x^n] A(x) * (4*n + 1 - F(4*x)) = 0 for n > 0.

a(n) = (n-1)! * Sum_{k=1..n} k! * 4^(k-1) * a(n-k) / (n-k)! for n > 0 with a(0) = 1.

Example

E.g.f.: A(x) = 1 + x + 9*x^2/2! + 217*x^3/3! + 10225*x^4/4! + 801681*x^5/5! + 94616761*x^6/6! + 15696482089*x^7/7! + ...
where the logarithm of A(x) equals the integer series
log(A(x)) = x + 4*x^2 + 32*x^3 + 384*x^4 + 6144*x^5 + 122880*x^6 + ... + (n-1)!*4^(n-1)*x^n + ...
Let G(x) be the g.f. of A158876, then
A(x)^4 = G(4*x) = 1 + (4*x) + 3*(4*x)^2/2! + 19*(4*x)^3/3! + 217*(4*x)^4/4! + 4041*(4*x)^5/5! + 113611*(4*x)^6/6! + ... + A158876(n)*4^n*x^n/n! + ...

Prog

(PARI) {a(n) = n! * polcoef( exp(sum(k=1, n, (k-1)! * 4^(k-1) * x^k) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k! * 4^(k-1) * a(n-k) / (n-k)!))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A158876, A393752, A393753, A393755, A393756, A393757, A393758.

Sequence in context: A197669 A368632 A371684 * A157692 A299548 A079913

Adjacent sequences: A393751 A393752 A393753 * A393755 A393756 A393757

Keyword

nonn

Author

Paul D. Hanna, Feb 27 2026

Status

approved