A207386 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1+x)/(1+x^3).

1, 1, 2, 6, 28, 172, 1269, 10879, 106343, 1167970, 14241792, 190919195, 2790920003, 44184957237, 753152722642, 13752229833566, 267809474619299, 5540559819166056, 121355678158129804, 2805498395990301867, 68265999939081386947, 1744058001878302097109

Offset: 0

Author

Paul D. Hanna, Feb 17 2012

Keywords

• nonn

Comments

Motivated by Peter Bala's identity described in 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),

here q = (1+x)/(1+x^3). See cross-references for other examples.

At present Bala's identity is conjectural and needs formal proof.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 28*x^4 + 172*x^5 + 1269*x^6 +...
Let q = (1+x)/(1+x^3) = 1/(1-x+x^2), then
A(x) = 1 + (q-1) + (q-1)*(q^2-1) + (q-1)*(q^2-1)*(q^3-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1)*(q^5-1) +...
which also is proposed to equal:
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 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250
• Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A158690, A158691, A179525, A207397, A207433.

Programs

PARI

{a(n)=local(A=1+x,q=(1+x)/(1+x^3 +x*O(x^n))); A=sum(m=0,n,prod(k=1,m,(q^k-1)));polcoeff(A,n)}

PARI

{a(n)=local(A=1+x,q=(1+x)/(1+x^3 +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=0,21,print1(a(n),", "))

Formula

G.f.: Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) where q = (1+x)/(1+x^3). [Based on Peter Bala's conjecture in A158690]

a(n) ~ n! * 2^(2*n+3/2) * 3^(n+1) / (exp(Pi^2/24) * Pi^(2*n+2)). - Vaclav Kotesovec, Aug 22 2017