A186737 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=1..n} ((1 + x*A(x))^k - 1).

1, 1, 3, 14, 82, 563, 4390, 38273, 370090, 3951524, 46436236, 598606795, 8431113021, 129134793211, 2139840753676, 38161937079832, 728817179712481, 14837170081079164, 320656534559643735, 7329984468062242588, 176662137501934534082, 4476371962949911027001, 118946746110478235868525

Offset

0,3

Comments

This is an example of Peter Bala's identity (cf. A158690):

Sum_{n>=0} Product_{k=1..n} (q^k - 1) = Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) at q = 1+x*A(x).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..115

Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], p. 33, 2019.

Formula

G.f. satisfies:

(1) A(x) = Sum_{n>=0} (1+x*A(x))^(-n^2) * Product_{k=1..n} ((1+x*A(x))^(2*k-1) - 1).

(2) A(x) = 1/(1 - (q-1)/(q - (q^2-1)/(q^2 - (q^3-1)/(q^3 - (q^4-1)/(q^4 - (q^5-1)/(q^5 -...)))))), a continued fraction, where q = 1+x*A(x).

a(n) ~ 6*sqrt(2) * exp(Pi^2/24) * 12^n * n! / Pi^(2*n+2). - Vaclav Kotesovec, May 05 2014

Example

G.f.: A(X) = 1 + x + 3*x^2 + 14*x^3 + 82*x^4 + 563*x^5 + 4390*x^6 +...
where
A(x) = 1 + ((1+x*A(x))-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1)*((1+x*A(x))^3-1) + ((1+x*A(x))-1)*((1+x*A(x))^2-1)*((1+x*A(x))^3-1)*((1+x*A(x))^4-1) +...
Let q = 1+x*A(x), then:
A(x) = 1 + (q-1)/q + (q-1)*(q^3-1)/q^4 + (q-1)*(q^3-1)*(q^5-1)/q^9 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)/q^16 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)*(q^9-1)/q^25 +...

Mathematica

nmax = 22; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Sum[Product[ ((1 + x A[x])^k - 1), {k, 1, j}], {j, 0, n}] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, prod(k=1, m, (1+x*A+x*O(x^n))^k - 1))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x, q); for(i=1, n, q=(1+x*A +x*O(x^n)); A=sum(m=0, n, q^(-m^2)*prod(k=1, m, (q^(2*k-1)-1)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A179525.

Sequence in context: A291535 A333956 A194091 * A367972 A020104 A220894

Adjacent sequences: A186734 A186735 A186736 * A186738 A186739 A186740

Keyword

nonn

Author

Paul D. Hanna, Apr 23 2013

Status

approved