A219535 G.f. satisfies A(x) = 1 + x*(2*A(x)^2 + A(x)^3).

1, 3, 21, 192, 2001, 22539, 267276, 3287496, 41556585, 536565225, 7046232285, 93820316412, 1263673602300, 17186898452772, 235709926636296, 3256050894487824, 45263067114496665, 632721425905230213, 8888476706476318047, 125418490224196533096, 1776734673565844413929

Offset

0,2

Links

Michael De Vlieger, Table of n, a(n) for n = 0..799

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

Formula

Let G(x) = (1-2*x - sqrt(1 - 8*x + 4*x^2)) / (2*x), then g.f. A(x) satisfies:

(1) A(x) = (1/x)*Series_Reversion(x/G(x)),

(2) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),

where x*G(x) is the g.f. of A047891.

Recurrence: 2*n*(2*n+1)*(11*n - 16)*a(n) = (649*n^3 - 1593*n^2 + 1130*n - 240)*a(n-1) + 16*(n-2)*(2*n-3)*(11*n-5)*a(n-2). - Vaclav Kotesovec, Dec 28 2013

a(n) ~ sqrt((33+17*sqrt(33))/11) * ((59+11*sqrt(33))/8)^n / (4 * sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 28 2013

From Seiichi Manyama, Jul 28 2020: (Start)

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n,k) * binomial(2*n+k+1,n)/(2*n+k+1).

a(n) = (1/(2*n+1)) * Sum_{k=0..n} 2^k * binomial(2*n+1,k) * binomial(3*n-k,n-k). (End)

From Seiichi Manyama, Aug 10 2023: (Start)

a(n) = (1/n) * Sum_{k=0..n-1} (-2)^k * 3^(n-k) * binomial(n,k) * binomial(3*n-k,n-1-k) for n > 0.

a(n) = (1/n) * Sum_{k=1..n} 3^k * binomial(n,k) * binomial(2*n,k-1) for n > 0. (End)

Example

G.f.: A(x) = 1 + 3*x + 21*x^2 + 192*x^3 + 2001*x^4 + 22539*x^5 +...
Related expansions:
A(x)^2 = 1 + 6*x + 51*x^2 + 510*x^3 + 5595*x^4 + 65148*x^5 +...
A(x)^3 = 1 + 9*x + 90*x^2 + 981*x^3 + 11349*x^4 + 136980*x^5 +...
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + 3*x + 12*x^2 + 57*x^3 + 300*x^4 + 1686*x^5 +...+ A047891(n+1)*x^n +...

Mathematica

CoefficientList[1/x*InverseSeries[Series[2*x^2/(1-2*x-Sqrt[1-8*x+4*x^2]), {x, 0, 21}], x], x] (* Vaclav Kotesovec, Dec 28 2013 *)

Prog

(PARI) /* Formula A(x) = 1 + x*(2*A(x)^2 + A(x)^3): */
{a(n)=my(A=1); for(i=1, n, A=1+x*(2*A^2+A^3) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Formula using Series Reversion: */
{a(n)=my(A=1, G=(1-2*x-sqrt(1-8*x+4*x^2+x^3*O(x^n)))/(2*x)); A=(1/x)*serreverse(x/G); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1)); \\ Seiichi Manyama, Jul 28 2020
(PARI) a(n) = sum(k=0, n, 2^k*binomial(2*n+1, k)*binomial(3*n-k, n-k))/(2*n+1); \\ Seiichi Manyama, Jul 28 2020

Crossrefs

Column k=2 of A336575.

Cf. A047891, A219534, A219536.

Sequence in context: A057616 A212072 A212029 * A292361 A369783 A151388

Adjacent sequences: A219532 A219533 A219534 * A219536 A219537 A219538

Keyword

nonn

Author

Paul D. Hanna, Nov 21 2012

Status

approved