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A242003

G.f. A(x) satisfies 2*A(x) = 1 + x + A(x*A(x)).

1, 1, 1, 3, 15, 101, 841, 8267, 93259, 1184693, 16718377, 259403303, 4389247891, 80446526037, 1587992497445, 33595010710967, 758426286470763, 18201458396436081, 462778682120158733, 12427549693656564655, 351513706699979429223, 10446113259707687607057

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OFFSET

0,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..330

FORMULA

a(n) ~ c * n^n / (exp(n) * (log(2))^n), where c = 1.16670181891916121...

G.f. satisfies A(x) = 1 + Sum_{n>=0} B^n(x) / 2^(n+1), where B(x) = x*A(x) and B^n(x) denotes the n-th iteration of B(x) with B^0(x) = x. - Paul D. Hanna, Apr 05 2026

From Seiichi Manyama, Apr 06 2026: (Start)

Let b(n,k) = [x^n] A(x)^k.

b(0,1) = b(1,1) = 1; b(n,1) = Sum_{j=1..n-1} b(j,1) * b(n-j,j).

For k > 1, b(0,k) = 1; b(n,k) = (1/n) * Sum_{j=1..n} ((k+1)*j-n) * b(j,1) * b(n-j,k).

a(n) = b(n,1). (End)

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 15*x^4 + 101*x^5 + 841*x^6 + 8267*x^7 + 93259*x^8 + 1184693*x^9 + 16718377*x^10 + ...

A(x*A(x)) = 1 + x + 2*x^2 + 6*x^3 + 30*x^4 + 202*x^5 + 1682*x^6 + 16534*x^7 + 186518*x^8 + ... where 2*A(x) = 1 + x + A(x*A(x)).

MATHEMATICA

nmax = 21; sol = {a[0] -> 1};

Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[2 A[x] - (1 + x + A[x A[x]]) + O[x]^(n+1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];

sol /. Rule -> Set;

a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1+x + subst(A, x, x*A +x*O(x^n)) - A); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Column k=1 of A392378.

Cf. A242004 (q=2), A242005 (q=3), A242006 (q=4).

Cf. A393888, A393889, A393890.

Cf. A006196.

Sequence in context: A074536 A152093 A109777 * A265164 A348793 A135903

Adjacent sequences: A242000 A242001 A242002 * A242004 A242005 A242006