A245108 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (A(x)^k + x^k)^k * x^(k*(n-k)).

1, 2, 5, 17, 89, 556, 3982, 31177, 261069, 2309922, 21435122, 207605687, 2091924418, 21883241424, 237301377573, 2664817940085, 30965904565393, 372114924929070, 4621600351308022, 59287502088944355, 785045224973677569, 10721533799643490991, 150898539708638128793, 2186680325880161158424

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..70

Formula

G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (1 + x^(n-k)*A(x)^k)^(n-k) * x^(k*(n-k)) * A(x)^(k^2).

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 17*x^3 + 89*x^4 + 556*x^5 + 3982*x^6 +...
Let A = A(x) then
A(x) = 1 + x*(1 + (A+x))
+ x^2*(1 + 2*(A+x)*x + (A^2+x^2)^2)
+ x^3*(1 + 3*(A+x)*x^2 + 3*(A^2+x^2)^2*x^2 + (A^3+x^3)^3)
+ x^4*(1 + 4*(A+x)*x^3 + 6*(A^2+x^2)^2*x^4 + 4*(A^3+x^3)^3*x^3 + (A^4+x^4)^4)
+ x^5*(1 + 5*(A+x)*x^4 + 10*(A^2+x^2)^2*x^6 + 10*(A^3+x^3)^3*x^6 + 5*(A^4+x^4)^4*x^4 + (A^5+x^5)^5)
+ x^6*(1 + 6*(A+x)*x^5 + 15*(A^2+x^2)^2*x^8 + 20*(A^3+x^3)^3*x^9 + 15*(A^4+x^4)^4*x^8 + 6*(A^5+x^5)^5*x^5 + (A^6+x^6)^6) +...
where we have the identity
A(x) = 1 + x*((1+x) + A)
+ x^2*((1+x^2)^2 + 2*(1+x*A)*x*A + A^4)
+ x^3*((1+x^3)^3 + 3*(1+x^2*A)^2*x^2*A + 3*(1+x*A^2)*x^2*A^4 + A^9)
+ x^4*((1+x^4)^4 + 4*(1+x^3*A)^3*x^3*A + 6*(1+x^2*A^2)^2*x^4*A^4 + 4*(1+x*A^3)*x^3*A^9 + A^16)
+ x^5*((1+x^5)^5 + 5*(1+x^4*A)^4*x^4*A + 10*(1+x^3*A^2)^3*x^6*A^4 + 10*(1+x^2*A^3)^2*x^6*A^9 + 5*(1+x*A^4)*x^4*A^16 + A^25)
+ x^6*((1+x^6)^6 + 6*(1+x^5*A)^5*x^5*A + 15*(1+x^4*A^2)^4*x^8*A^4 + 20*(1+x^3*A^3)^3*x^9*A^9 + 15*(1+x^2*A^4)^2*x^8*A^16 + 6*(1+x*A^5)*x^5*A^25 + A^36) +...

Prog

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

Crossrefs

Cf. A245107.

Sequence in context: A162043 A016121 A216519 * A195137 A026822 A059248

Adjacent sequences: A245105 A245106 A245107 * A245109 A245110 A245111

Keyword

nonn

Author

Paul D. Hanna, Jul 24 2014

Status

approved