A209357 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^(n+1)*A(x)) / (1 - x^n).

1, 1, 3, 6, 14, 31, 72, 166, 390, 922, 2197, 5273, 12728, 30892, 75327, 184476, 453505, 1118798, 2768843, 6872437, 17103411, 42670102, 106697009, 267359854, 671260241, 1688411587, 4254084396, 10735614274, 27132998096, 68671994940, 174035109012, 441607820562

Offset: 0

Author

Paul D. Hanna, Mar 06 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 14*x^4 + 31*x^5 + 72*x^6 + 166*x^7 +...
where the g.f. satisfies the identity:
A(x) = (1+x^2*A(x))/(1-x) * (1+x^3*A(x))/(1-x^2) * (1+x^4*A(x))/(1-x^3) *...
A(x) = 1 + x*(1+x*A(x))/(1-x) + x^2*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)) + x^3*(1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))/((1-x)*(1-x^2)*(1-x^3)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..455

Crossrefs

Cf. A190861, A190862.

Programs

Mathematica

(* Calculation of constants {d,c}: *) Chop[{1/r, Sqrt[(s*(1 + r*s)*Log[r]*(s*(1 + r*s)*(-QPochhammer[r]*(Log[1 - r] + Log[r] + QPolyGamma[0, 1, r]) + r*Log[r]*Derivative[0, 1][QPochhammer][r, r]) - r*Log[r]*Derivative[0, 1][QPochhammer][-r*s, r])) / (2*Pi*QPochhammer[r] * (r*s*Log[r]^2 + (1 + r*s)^2*QPolyGamma[1, Log[-r*s]/Log[r], r]))]} /. FindRoot[{s*(1 + r*s) == QPochhammer[-r*s, r]/QPochhammer[r], Log[1-r] + r*s*Log[r]/(1 + r*s) + QPolyGamma[0, Log[-r*s]/Log[r], r] == -Log[r]}, {r, 2/5}, {s, 2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 10 2025 *)

PARI

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

PARI

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*prod(k=1, m, (1+x^k*A)/(1-x^k+x*O(x^n))))); polcoeff(A, n)}
for(n=0,35,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = Sum_{n>=0} x^n*Product_{k=1..n} (1 + x^k*A(x))/(1-x^k) due to the q-binomial theorem.

a(n) ~ c * d^n / n^(3/2), where d = 2.6481816651621274063587047915... and c = 7.257947883786923940523402074... - Vaclav Kotesovec, Jun 10 2025