A218222 G.f. A(x) satisfies: A(x) = x + x*[d/dx A(x)^2].

1, 2, 12, 112, 1360, 19872, 335104, 6359040, 133560576, 3069007360, 76493880320, 2054400577536, 59136549994496, 1816392567062528, 59305340822814720, 2051451257317490688, 74958908119819812864, 2885480280276224311296, 116731741304854533111808

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Formula

a(n) = 2^(n-1)*A088716(n-1) for n>=1, where g.f. F(x) of A088716 satisfies: F(x) = 1 + x*F(x)*d/dx[x*F(x)].

a(n) = n*A112915(n-1) for n>=1.

G.f.: x*d/dx x*G(x), where g.f. G(x) of A112915 satisfies: G(x) = 1 + x*(d/dx[x*G(x)])^2.

a(n) ~ c * n * 2^(n-1) * n!, where c = A238223 = 0.21795078944715106549... - Vaclav Kotesovec, Aug 24 2017

Example

G.f.: A(x) = x + 2*x^2 + 12*x^3 + 112*x^4 + 1360*x^5 + 19872*x^6 +...
Related series:
A(x)^2 = x^2 + 4*x^3 + 28*x^4 + 272*x^5 + 3312*x^6 + 47872*x^7 + 794880*x^8 + 14840064*x^9 +...+ A112915(n-1)*x^n +...
d/dx A(x)^2 = 2*x + 12*x^2 + 112*x^3 + 1360*x^4 + 19872*x^5 +...

Maple

a:= proc(n) option remember; `if`(n<2, 1,
      n*add(a(i)*a(n-i), i=1..n-1))
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 05 2020

Mathematica

a[n_] := a[n] = If[n<2, 1, n*Sum[a[i]*a[n-i], {i, 1, n-1}]];
Array[a, 20] (* Jean-François Alcover, Dec 18 2020, after Maple *)

Prog

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

Crossrefs

Cf. A000699, A112915, A185183.

Sequence in context: A199045 A009232 A349311 * A355112 A374562 A292187

Adjacent sequences: A218219 A218220 A218221 * A218223 A218224 A218225

Keyword

nonn

Author

Paul D. Hanna, Jan 31 2013

Status

approved