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A226761 G.f.: 1 / (1 + 12*x*G(x)^2 - 13*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

%I A226761 #31 Aug 15 2025 12:18:51
%S A226761 1,1,16,118,1004,8601,75076,662796,5903676,52949332,477533356,
%T A226761 4326309406,39343725716,358943047438,3283745710968,30112624408488,
%U A226761 276715616909148,2547523969430508,23491659440021920,216942761366305144,2006084011596742384,18572529488934397689
%N A226761 G.f.: 1 / (1 + 12*x*G(x)^2 - 13*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%H A226761 Vincenzo Librandi and Joerg Arndt, <a href="/A226761/b226761.txt">Table of n, a(n) for n = 0..200</a>
%F A226761 a(n) = Sum_{k=0..n} C(3*k, n-k) * C(4*n-3*k, k).
%F A226761 a(n) = Sum_{k=0..n} C(n+3*k, n-k) * C(3*n-3*k, k).
%F A226761 a(n) = Sum_{k=0..n} C(2*n+3*k, n-k) * C(2*n-3*k, k).
%F A226761 a(n) = Sum_{k=0..n} C(3*n+3*k, n-k) * C(n-3*k, k).
%F A226761 a(n) = Sum_{k=0..n} C(4*n+3*k, n-k) * C(-3*k, k).
%F A226761 G.f.: 1 / (1 - x*G(x)^2 - 13*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%F A226761 a(n) ~ 2^(8*n+5/2)/(7*3^(3*n+1/2)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jun 17 2013
%F A226761 From _Seiichi Manyama_, Aug 05 2025: (Start)
%F A226761 a(n) = [x^n] 1/((1+3*x) * (1-x)^(3*n+1)).
%F A226761 a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(4*n+1,k).
%F A226761 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n+k,k). (End)
%F A226761 From _Seiichi Manyama_, Aug 14 2025: (Start)
%F A226761 a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
%F A226761 G.f.: G(x)^2/((-3+4*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
%F A226761 G.f.: B(x)^2/(1 + 7*(B(x)-1)/4), where B(x) is the g.f. of A005810. - _Seiichi Manyama_, Aug 15 2025
%e A226761 G.f.: A(x) = 1 + x + 16*x^2 + 118*x^3 + 1004*x^4 + 8601*x^5 +...
%e A226761 A related series is G(x) = 1 + x*G(x)^4, where
%e A226761 G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%e A226761 G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
%e A226761 G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
%e A226761 such that A(x) = 1/(1 + 12*x*G(x)^2 - 13*x*G(x)^3).
%t A226761 Table[Sum[Binomial[2*n+3*k,n-k]*Binomial[2*n-3*k,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 17 2013 *)
%o A226761 (PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+12*x*G^2-13*x*G^3), n)}
%o A226761 for(n=0, 30, print1(a(n), ", "))
%o A226761 (PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-x*G^2-13*x^2*G^6), n)}
%o A226761 for(n=0, 30, print1(a(n), ", "))
%o A226761 (PARI) {a(n)=sum(k=0, n, binomial(2*n+3*k, n-k)*binomial(2*n-3*k, k))}
%o A226761 for(n=0, 30, print1(a(n), ", "))
%o A226761 (PARI) {a(n)=sum(k=0, n, binomial(3*k, n-k)*binomial(4*n-3*k, k))}
%o A226761 for(n=0, 30, print1(a(n), ", "))
%o A226761 (PARI) {a(n)=sum(k=0, n, binomial(4*n+3*k, n-k)*binomial(-3*k, k))}
%o A226761 for(n=0, 30, print1(a(n), ", "))
%Y A226761 Cf. A002293.
%Y A226761 Cf. A005810, A078995, A147855, A226733, A385605, A386811.
%K A226761 nonn
%O A226761 0,3
%A A226761 _Paul D. Hanna_, Jun 16 2013