A218224 -id:A218224 - cp's OEIS frontend

A185183 G.f. A(x) satisfies A(x) = 1+x + x^2*[d/dx A(x)^2].

1, 1, 2, 10, 72, 672, 7640, 102072, 1564864, 27064448, 521248320, 11064781760, 256702399360, 6462978471168, 175520877380992, 5115062135795584, 159227683153536000, 5273353734210310144, 185143079148664099840, 6869062513111759635456

Offset: 0

Paul D. Hanna, Mar 12 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 72*x^4 + 672*x^5 + 7640*x^6 +...
Related series:
A(x)^2 = 1 + 2*x + 5*x^2 + 24*x^3 + 168*x^4 + 1528*x^5 + 17012*x^6 +...
d/dx A(x)^2 = 2 + 10*x + 72*x^2 + 672*x^3 + 7640*x^4 + 102072*x^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A143916, A218223, A218224, A376125.

{a(n)=local(A=1+x); for(i=1, n, A=1+x+x^2*deriv(A^2+x*O(x^n))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies: A(x) = 1+x + 2*x^2*A(x)*A'(x).
a(n) ~ c * n! * 2^n / sqrt(n), where c = 0.493602095524198015213766719826126125048... - Vaclav Kotesovec, Feb 21 2014
a(0) = a(1) = 1; a(n) = (n-1) * Sum_{k=0..n-1} a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 05 2020
a(0) = a(1) = 1; a(n) = 2 * Sum_{k=0..n-1} k * a(k) * a(n-1-k). - Seiichi Manyama, Jul 16 2025

A218223 G.f. A(x) satisfies: A(x) = 1+x + x^2*[d/dx A(x)^3].

Original entry on oeis.org

1, 1, 3, 24, 273, 3996, 70785, 1465506, 34662222, 921511944, 27201024639, 882828325530, 31253560065684, 1198758613494852, 49530067909218819, 2193498057583259784, 103664556373964098860, 5207896547115772335552, 277161367378578537506868

Offset: 0

Paul D. Hanna, Oct 23 2012

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 273*x^4 + 3996*x^5 + 70785*x^6 +...
Related series:
A(x)^3 = 1 + 3*x + 12*x^2 + 91*x^3 + 999*x^4 + 14157*x^5 + 244251*x^6 +...
d/dx A(x)^3 = 3 + 24*x + 273*x^2 + 3996*x^3 + 70785*x^4 + 1465506*x^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..320

Cf. A185183, A218224.

{a(n)=local(A=1+x); for(i=1, n, A=1+x+x^2*deriv(A^3+x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) satisfies: A(x) = 1+x + 3*x^2*A(x)^2*A'(x).

a(n) ~ c * 3^n * n! / n^(1/3), where c = 0.3133426736012301024021... - Vaclav Kotesovec, Aug 24 2017

cp's OEIS Frontend

A185183 G.f. A(x) satisfies A(x) = 1+x + x^2*[d/dx A(x)^2].

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