_Paul D. Hanna (pauldhanna(AT)juno.com), _, Nov 11 2011

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proposed

G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k] * x^n*A(x)^n/n ).

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G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k] * x^n*A(x)^n/n ).

G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A198950.

G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 51*x^4 + 233*x^5 + 1126*x^6 +...

log(A(x)) = (1 + x)*x*A(x) + (1 + 2^3*x) + x^2)*x^2*A(x)^2/2 +

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

Cf. A198950.

allocated for Paul D. Hanna

G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k] * x^n*A(x)^n/n ).

1, 1, 3, 12, 51, 233, 1126, 5639, 29001, 152364, 814268, 4412733, 24193883, 133964582, 748084930, 4208233719, 23825366565, 135657095301, 776310257573, 4462597387720, 25757503356976, 149216036734018, 867316809868018, 5056666312351966, 29564095191382323

0,3

G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A198950.

G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 51*x^4 + 233*x^5 + 1126*x^6 +...

where

log(A(x)) = (1 + x)*x*A(x) + (1 + 2^3*x) + x^2)*x^2*A(x)^2/2 +

(1 + 3^3*x + 3^3*x^2 + x^3)*x^3*A(x)^3/3 +

(1 + 4^3*x + 6^3*x^2 + 4^3*x^3 + x^4)*x^4*A(x)^4/4 +

(1 + 5^3*x + 10^3*x^2 + 10^3*x^3 + 5^3*x^4 + x^5)*x^5*A(x)^5/5 +...

more explicitly,

log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 149*x^4/4 + 821*x^5/5 + 4664*x^6/6 + 26839*x^7/7 +...

The g.f. satisfies A(x) = G(x*A(x)) where G(x) is the g.f. of A198950:

G(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 22*x^5 + 58*x^6 + 150*x^7 + 392*x^8 +...

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

Cf. A198950.

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Paul D. Hanna (pauldhanna(AT)juno.com), Nov 11 2011

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