A213282 G.f. satisfies A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

1, 1, 6, 36, 236, 1656, 12192, 92960, 727824, 5817696, 47281472, 389533056, 3245867136, 27308274688, 231654031104, 1979205694464, 17016094611712, 147104972637696, 1277988764697600, 11151534242977792, 97692088569096192, 858890594909048832, 7575804347863105536

Offset

0,3

Comments

Compare to the g.f. B(x) of A006319 where B(x) = C(x/(1-x)^2) such that C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f. satisfies: A(x) = F(x*A(x)^3) where F(x) = 1 + x/F(-x)^3 is the g.f. of A213281.

G.f. A(x) satisfies: A(1 - G(-x)) = G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * binomial(3*k,k)/(2*k+1). - Seiichi Manyama, Oct 03 2023

Example

G.f.: A(x) = 1 + x + 6*x^2 + 36*x^3 + 236*x^4 + 1656*x^5 + 12192*x^6 +...
G.f.: A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3 is g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

Maple

series(RootOf(G = 1 + G^3*x/(1-x)^3, G), x=0, 30); # Mark van Hoeij, Apr 18 2013

Prog

(PARI) /* G.f. A(x) = G(x/(1-x)^3) where G(x) = 1 + x*G(x)^3: */
{a(n)=local(A, G=1+x); for(i=1, n, G=1+x*G^3+x*O(x^n)); A=subst(G, x, x/(1-x+x*O(x^n))^3); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* G.f. A(x) = F(x*A(x)^3) where F(x) = 1 + x/F(-x)^3: */
{a(n)=local(F=1+x+x*O(x^n), A=1); for(i=1, n+1, F=1+x/subst(F^3, x, -x+x*O(x^n))); A=(serreverse(x/F^3)/x)^(1/3); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A213281, A001764; variants: A006319 (royal paths in a lattice), A213336.

Sequence in context: A391202 A057395 A259819 * A066053 A344250 A153824

Adjacent sequences: A213279 A213280 A213281 * A213283 A213284 A213285

Keyword

nonn

Author

Paul D. Hanna, Jun 08 2012

Status

approved