A140097 G.f. satisfies: A(x) = Series_Reversion( x/(1 + A(x) + A(x)^2) ).

1, 1, 3, 12, 60, 346, 2218, 15444, 115075, 908155, 7535185, 65374018, 590579575, 5537249212, 53742567000, 538801229874, 5570060420573, 59288164937748, 648934780013375, 7295904025820975, 84174136470742517, 995682428049720830

Offset

1,3

Links

Table of n, a(n) for n=1..22.

Formula

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

a(n)=T(n,1), T(n,j)=-sum(m=j..n-1, T(m,j)*sum(i=1..n-m, (sum(k=1..i, (-1)^k*binomial(k,i-k)*binomial(m+k-1,m-1)))*T(n-m,i))), n>j, T(n,n)=1. [Vladimir Kruchinin, May 04 2012]

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 346*x^6 + 2218*x^7 +...
A(A(x)) = x + 2*x^2 + 8*x^3 + 40*x^4 + 234*x^5 + 1526*x^6 +10816*x^7+...
A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 112*x^5 + 692*x^6 + 4628*x^7 +...
x = A(x*[1 - A(x) + 2*A(x)^2 - 4*A(x)^3 + 9*A(x)^4 - 21*A(x)^5 +-...]).

Prog

(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x/(1+A+A^2 +x*O(x^n)))); polcoeff(A, n))}
(Maxima) T(n, j):=if n=j then 1 else -sum(T(m, j)*sum((sum((-1)^k*binomial(k, i-k)*binomial(m+k-1, m-1), k, 1, i))*T(n-m, i), i, 1, n-m), m, j, n-1); makelist(T(n, 1), n, 1, 10); [Vladimir Kruchinin, May 04 2012]

Crossrefs

Cf. A001006 (Motzkin numbers).

Sequence in context: A181282 A020052 A096471 * A105227 A370652 A000258

Adjacent sequences: A140094 A140095 A140096 * A140098 A140099 A140100

Keyword

nonn

Author

Paul D. Hanna, May 15 2008

Status

approved