A226751 - OEIS

%I #29 Aug 15 2025 15:58:10

%S 1,1,9,48,289,1761,10932,68664,435201,2777763,17829489,114968052,

%T 744178716,4832624044,31469746632,205422018288,1343734578561,

%U 8806130111847,57805893969531,380013533789928,2501507255441049,16486378106441697,108773240389894056

%N G.f.: 1 / (1 + 6*x*G(x) - 7*x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

%H Vincenzo Librandi and Joerg Arndt, <a href="/A226751/b226751.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{k=0..n} C(2*k, n-k) * C(3*n-2*k, k).

%F a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(2*n-2*k, k).

%F a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(n-2*k, k).

%F a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(-2*k, k).

%F G.f.: 1/(1 - x*G(x) - 7*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

%F a(n) ~ 3^(3*n+3/2)/(5*sqrt(Pi*n)*2^(2*n+1)). - _Vaclav Kotesovec_, Jun 17 2013

%F Conjecture: 18*n*(2*n-1)*(55*n-76)*a(n) +(-11605*n^3+28521*n^2-20870*n+4536)*a(n-1) -24*(55*n-21)*(3*n-4)*(3*n-2)*a(n-2)=0. - _R. J. Mathar_, Jun 14 2016

%F From _Seiichi Manyama_, Aug 05 2025: (Start)

%F a(n) = [x^n] 1/((1+2*x) * (1-x)^(2*n+1)).

%F a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n+1,k).

%F a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(2*n+k,k). (End)

%F From _Seiichi Manyama_, Aug 14 2025: (Start)

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

%F G.f.: G(x)^2/((-2+3*G(x)) * (3-2*G(x))) where G(x) = 1+x*G(x)^3 is the g.f. of A001764. (End)

%F G.f.: B(x)^2/(1 + 5*(B(x)-1)/3), where B(x) is the g.f. of A005809. - _Seiichi Manyama_, Aug 15 2025

%e G.f.: A(x) = 1 + x + 9*x^2 + 48*x^3 + 289*x^4 + 1761*x^5 + 10932*x^6 +...

%e A related series is G(x) = 1 + x*G(x)^3, where

%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

%e G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...

%e such that A(x) = 1/(1 + 6*x*G(x) - 7*x*G(x)^2).

%t Table[Sum[Binomial[n+2*k,n-k]*Binomial[2*n-2*k,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 17 2013 *)

%o (PARI) {a(n)=sum(k=0, n, binomial(n+2*k, n-k)*binomial(2*n-2*k, k))}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(3*n-2*k, k))}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^3+x*O(x^n)); polcoeff(1/(1+6*x*G-7*x*G^2), n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^3+x*O(x^n)); polcoeff(1/(1-x*G-7*x^2*G^4), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A147855, A001764.

%Y Cf. A005809, A006256, A160906, A183160.

%Y Cf. A226705, A226733.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 16 2013