A214692 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(4*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

1, 1, 2, 11, 71, 515, 3997, 32488, 273009, 2352724, 20678966, 184660333, 1670619561, 15279692008, 141048655988, 1312429249996, 12296515232446, 115909188223053, 1098444610424929, 10459429664510189, 100021237512559055, 960168745226226195, 9249466125601138425

Offset

0,3

Comments

Compare the g.f. to the identity:

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

which holds for all power series G(x) such that G(0)=1.

Links

G. C. Greubel, Table of n, a(n) for n = 0..980

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Formula

G.f. A(x) satisfies:

(1) 1+x = A(y) where y = x - 2*x^2 - 3*x^3 - x^4, which is the g.f. of row 2 in triangle A214690.

(2) x = Sum_{n>=1} 1/A(x)^(n*(n+4)) * Product_{k=1..n} (A(x)^(2*k-1) - 1).

(3) 1+x = A(x)^2 + A(x)^3 - A(x)^4. - Paul D. Hanna, Nov 15 2014

a(n) ~ sqrt(145/sqrt(41)-21) * ((213+41*sqrt(41))/46)^n / (16 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 29 2014

Example

G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 71*x^4 + 515*x^5 + 3997*x^6 +...
The g.f. satisfies:
x = (A(x)-1)/A(x)^5 + (A(x)-1)*(A(x)^3-1)/A(x)^12 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)/A(x)^21 + (A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)/A(x)^32 +
(A(x)-1)*(A(x)^3-1)*(A(x)^5-1)*(A(x)^7-1)*(A(x)^9-1)/A(x)^45 +...
Related expansions.
A(x)^2 = 1 + 2*x + 5*x^2 + 26*x^3 + 168*x^4 + 1216*x^5 + 9429*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 46*x^3 + 297*x^4 + 2148*x^5 + 16649*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 72*x^3 + 465*x^4 + 3364*x^5 + 26078*x^6 +...
where 1+x = A(x)^2 + A(x)^3 - A(x)^4.

Mathematica

CoefficientList[1+InverseSeries[Series[x - 2*x^2 - 3*x^3 - x^4, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 29 2014 *)

Prog

(PARI) {a(n)=if(n<0, 0, polcoeff(1 + serreverse(x - 2*x^2 - 3*x^3 - x^4 +x^2*O(x^n)), n))}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-polcoeff(sum(m=1, #A, 1/Ser(A)^(4*m)*prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* From 1+x = A(x)^2 + A(x)^3 - A(x)^4: */
{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^2+Ser(A)^3-Ser(A)^4)[#A]); A[n+1]}
for(n=0, 25, print1(a(n) , ", "))

Crossrefs

Cf. A214690, A214693, A214694, A214695, A001002 (variant).

Sequence in context: A371577 A334048 A047776 * A186633 A291301 A154887

Adjacent sequences: A214689 A214690 A214691 * A214693 A214694 A214695

Keyword

nonn

Author

Paul D. Hanna, Jul 26 2012

Status

approved