A301433 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * (1 + (1+x)^n * A(x))^n.

1, 2, 7, 33, 180, 1071, 6776, 44907, 308903, 2192218, 15986187, 119458754, 913085921, 7130596660, 56856959415, 462789536503, 3845600755712, 32634320897898, 282973974102022, 2508805646347474, 22758825619762568, 211400392340768128, 2011934295358876840, 19628847839908163287, 196377194100036473222, 2014897715206819032244, 21199978342486061320347, 228666292563823622317191

Offset

0,2

Links

Table of n, a(n) for n=0..27.

Formula

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

Example

G.f.: A(x) = 1 + 2*x + 7*x^2 + 33*x^3 + 180*x^4 + 1071*x^5 + 6776*x^6 + 44907*x^7 + 308903*x^8 + 2192218*x^9 + 15986187*x^10 + ...
such that
A(x) = 1 + x*(1 + (1+x)*A(x)) + x^2*(1 + (1+x)^2*A(x))^2 + x^3*(1 + (1+x)^3*A(x))^3 + x^4*(1 + (1+x)^4*A(x))^4 + x^5*(1 + (1+x)^5*A(x))^5 + x^6*(1 + (1+x)^6*A(x))^6 + ...
Also,
A(x) = 1/(1-x)  +  x*A(x)*(1+x)/(1 - x*(1+x))^2  +  x^2*A(x)^2*(1+x)^4/(1 - x*(1+x)^2)^3  +  x^3*A(x)^3*(1+x)^9/(1 - x*(1+x)^3)^4  +  x^4*A(x)^4*(1+x)^16/(1 - x*(1+x)^4)^5  + ...

Prog

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

Crossrefs

Cf. A302061, A301434.

Sequence in context: A369269 A369300 A055724 * A054727 A086618 A224769

Adjacent sequences: A301430 A301431 A301432 * A301434 A301435 A301436

Keyword

nonn

Author

Paul D. Hanna, Mar 25 2018

Status

approved