A301388 G.f. A(x) satisfies: A(x) = x * (1 + x*A'(x)) / (1 - x*A'(x)).

1, 2, 10, 78, 802, 10058, 147442, 2461054, 45960098, 948268402, 21410711450, 525049525294, 13897732641954, 394987936658714, 11999174713271266, 388077151776127486, 13315213471551257154, 483131189591348032482, 18485324379022683692714, 743888762544523242047886, 31411035576323146658185122, 1388712621964856674998780010, 64156199255423145619052883154, 3091505922381615544789816776830

Offset

1,2

Comments

O.g.f. equals the logarithm of the e.g.f. of A301387.

The e.g.f. G(x) of A301387 satisfies: [x^n] G(x)^n = (2*n - 1) * [x^(n-1)] G(x)^n for n>=1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

O.g.f. A(x) satisfies: [x^n] exp( n * A(x) ) = (2*n - 1) * [x^(n-1)] exp( n * A(x) ) for n>=1.

a(n) ~ c * 2^n * n!, where c = 0.321697697353832218399635... - Vaclav Kotesovec, Mar 21 2018

Example

G.f.: A(x) = x + 2*x^2 + 10*x^3 + 78*x^4 + 802*x^5 + 10058*x^6 + 147442*x^7 + 2461054*x^8 + 45960098*x^9 + 948268402*x^10 + ...
where
A(x) = x*(1 + x*A'(x)) / (1 - x*A'(x)).
RELATED SERIES.
A'(x) = 1 + 4*x + 30*x^2 + 312*x^3 + 4010*x^4 + 60348*x^5 + 1032094*x^6 + 19688432*x^7 + 413640882*x^8 + ...
exp(A(x)) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2185*x^4/4! + 108881*x^5/5! + 8012941*x^6/6! + 809101945*x^7/7! + 106751544593*x^8/8! + ... + A301387*x^n/n! + ...

Prog

(PARI) {a(n) = my(A=x); for(i=0, n, A = x*(1 + x*A')/(1 - x*A' +x*O(x^n)) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A301387.

Sequence in context: A381521 A307722 A138273 * A052568 A063170 A375878

Adjacent sequences: A301385 A301386 A301387 * A301389 A301390 A301391

Keyword

nonn

Author

Paul D. Hanna, Mar 20 2018

Status

approved