A153296 G.f.: A(x) = F(x*G(x)^3) = F(G(x)-1) where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.

1, 1, 5, 29, 180, 1162, 7698, 51950, 355531, 2460224, 17178755, 120861710, 855828960, 6094211829, 43610311298, 313449094851, 2261820356684, 16379528485200, 119003715014955, 867198605427231, 6336861345197670

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1119

Formula

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

G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^2 where G(x) is the g.f. of A001764.

G.f. satisfies: A(x/F(x)) = F(x*F(x)^2) where F(x) is the g.f. of A000108.

D-finite with recurrence: -6414187500*n*(3*n + 5)*(3*n + 2)*(3*n + 4)*(3*n + 1)*a(n) + 23625*(3*n + 5)*(3*n + 4)*(1731073*n^3 + 8176055*n^2 + 12702078*n + 6333840)*a(n + 1) - 90*(1336167457*n^5 + 16094421018*n^4 + 78187430279*n^3 + 191378559762*n^2 + 235603587684*n + 116458335840)*a(n + 2) + 36*(638707173*n^5 + 10523255220*n^4 + 69525306355*n^3 + 230296919380*n^2 + 382403293232*n + 254564421440)*a(n + 3) - 24*(109418234*n^5 + 2262905826*n^4 + 18692981906*n^3 + 77142386421*n^2 + 159132429377*n + 131341041300)*a(n + 4) + 48*(3446372*n^5 + 85215384*n^4 + 840822552*n^3 + 4140222927*n^2 + 10178021740*n + 9997627728)*a(n + 5) - 224*(n + 7)*(20872*n^4 + 459296*n^3 + 3786134*n^2 + 13858831*n + 19009992)*a(n + 6) + 896*(4*n + 25)*(n + 8)*(n + 7)*(4*n + 27)*(2*n + 13)*a(n + 7) = 0. - Robert Israel, Mar 18 2026

a(n) ~ 5^(3*n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(4*n + 1/2)). - Vaclav Kotesovec, Mar 18 2026

Example

G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 180*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*x^4 + 2974*x^5 +...
G(x)^3*A(x)^2 = 1 + 5*x + 29*x^2 + 180*x^3 + 1162*x^4 +...

Maple

f:= gfun:-rectoproc({-6414187500*n*(3*n + 5)*(3*n + 2)*(3*n + 4)*(3*n + 1)*a(n) + 23625*(3*n + 5)*(3*n + 4)*(1731073*n^3 + 8176055*n^2 + 12702078*n + 6333840)*a(n + 1) - 90*(1336167457*n^5 + 16094421018*n^4 + 78187430279*n^3 + 191378559762*n^2 + 235603587684*n + 116458335840)*a(n + 2) + 36*(638707173*n^5 + 10523255220*n^4 + 69525306355*n^3 + 230296919380*n^2 + 382403293232*n + 254564421440)*a(n + 3) - 24*(109418234*n^5 + 2262905826*n^4 + 18692981906*n^3 + 77142386421*n^2 + 159132429377*n + 131341041300)*a(n + 4) + 48*(3446372*n^5 + 85215384*n^4 + 840822552*n^3 + 4140222927*n^2 + 10178021740*n + 9997627728)*a(n + 5) - 224*(n + 7)*(20872*n^4 + 459296*n^3 + 3786134*n^2 + 13858831*n + 19009992)*a(n + 6) + 896*(4*n + 25)*(n + 8)*(n + 7)*(4*n + 27)*(2*n + 13)*a(n + 7), a(0) = 1, a(1) = 1, a(2) = 5, a(3) = 29, a(4) = 180, a(5) = 1162, a(6) = 7698}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 18 2026

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(2*k+1, k)/(2*k+1)*binomial(3*(n-k)+3*k, n-k)*3*k/(3*(n-k)+3*k)))}

Crossrefs

Cf. A000108, A001764; A153295, A153297.

Sequence in context: A190802 A139174 A290117 * A194723 A190917 A153391

Adjacent sequences: A153293 A153294 A153295 * A153297 A153298 A153299

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved