A393758 E.g.f.: exp( x * Sum_{n>=0} n! * 8^n * x^n ).

1, 1, 17, 817, 77665, 12299681, 2927458801, 977470825297, 435879178173377, 250203384739543105, 179679913060513513681, 157803532571496752836721, 166379813794981914558563617, 207381133084068630121614737377, 301636151761882780863343070164145, 506313774511002681383380409639226001

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(8*x) ).

(2) A(x) = exp(x + 8*x^2 * A'(x)/A(x)).

(3) A(x) = G(8*x)^(1/8) where G(x) is the g.f. of A158876.

(4) A'(x)/A(x) = Sum_{n>=0} (n+1)! * 8^n * x^n.

(5) [x^n] A(x)^(8*n) * (2 - F(8*x)) = 0 for n > 0.

(6) [x^n] A(x) * (8*n + 1 - F(8*x)) = 0 for n > 0.

a(n) = (n-1)! * Sum_{k=1..n} k! * 8^(k-1) * a(n-k) / (n-k)! for n > 0 with a(0) = 1.

Example

E.g.f.: A(x) = 1 + x + 17*x^2/2! + 817*x^3/3! + 77665*x^4/4! + 12299681*x^5/5! + 2927458801*x^6/6! + 977470825297*x^7/7! + ...
where the logarithm of A(x) equals the integer series
log(A(x)) = x + 8*x^2 + 128*x^3 + 3072*x^4 + 98304*x^5 + 3932160*x^6 + ... + (n-1)!*8^(n-1)*x^n + ...
Let G(x) be the g.f. of A158876, then
A(x)^8 = G(8*x) = 1 + (8*x) + 3*(8*x)^2/2! + 19*(8*x)^3/3! + 217*(8*x)^4/4! + 4041*(8*x)^5/5! + 113611*(8*x)^6/6! + ... + A158876(n)*8^n*x^n/n! + ...

Prog

(PARI) {a(n) = n! * polcoef( exp(sum(k=1, n, (k-1)! * 8^(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! * 8^(k-1) * a(n-k) / (n-k)!))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A158876, A393752, A393753, A393754, A393755, A393756, A393757.

Sequence in context: A249459 A191963 A328138 * A351181 A351769 A139091

Adjacent sequences: A393755 A393756 A393757 * A393759 A393760 A393761

Keyword

nonn

Author

Paul D. Hanna, Feb 27 2026

Status

approved