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A226733 G.f.: 1 / (1 + 8*x*G(x)^2 - 10*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

%I A226733 #37 Aug 15 2025 15:58:26
%S A226733 1,2,18,142,1186,10152,88414,779508,6936066,62159224,560238728,
%T A226733 5072970366,46114086446,420558296888,3846232573236,35261290343112,
%U A226733 323952686556354,2981787128165592,27491128592627800,253835886034173848,2346892194318851016,21724880414632781472
%N A226733 G.f.: 1 / (1 + 8*x*G(x)^2 - 10*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%H A226733 Vincenzo Librandi, <a href="/A226733/b226733.txt">Table of n, a(n) for n = 0..200</a>
%F A226733 a(n) = Sum_{k=0..n} C(2*k, n-k) * C(4*n-2*k, k).
%F A226733 a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(3*n-2*k, k).
%F A226733 a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(2*n-2*k, k).
%F A226733 a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(n-2*k, k).
%F A226733 a(n) = Sum_{k=0..n} C(4*n+2*k, n-k) * C(-2*k, k).
%F A226733 G.f.: 1 / (1 - 2*x*G(x)^2 - 10*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%F A226733 a(n) ~ 2^(8*n+3/2)/(3^(3*n+3/2)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jun 16 2013
%F A226733 From _Seiichi Manyama_, Aug 05 2025: (Start)
%F A226733 a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n+1)).
%F A226733 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n+1,k).
%F A226733 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k,k). (End)
%F A226733 From _Seiichi Manyama_, Aug 14 2025: (Start)
%F A226733 a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).
%F A226733 G.f.: G(x)^2/((-2+3*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)
%F A226733 G.f.: B(x)^2/(1 + 3*(B(x)-1)/2), where B(x) is the g.f. of A005810. - _Seiichi Manyama_, Aug 15 2025
%e A226733 G.f.: A(x) = 1 + 2*x + 18*x^2 + 142*x^3 + 1186*x^4 + 10152*x^5 +...
%e A226733 A related series is G(x) = 1 + x*G(x)^4, where
%e A226733 G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%e A226733 G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
%e A226733 G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
%e A226733 such that A(x) = 1/(1 + 8*x*G(x)^2 - 10*x*G(x)^3).
%t A226733 Table[Sum[Binomial[2*n+2*k,n-k]*Binomial[2*n-2*k,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 16 2013 *)
%o A226733 (PARI) {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+8*x*G^2-10*x*G^3), n)}
%o A226733 for(n=0, 30, print1(a(n), ", "))
%o A226733 (PARI) {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-2*x*G^2-10*x^2*G^6), n)}
%o A226733 for(n=0, 30, print1(a(n), ", "))
%o A226733 (PARI) {a(n)=sum(k=0, n, binomial(2*n+2*k, n-k)*binomial(2*n-2*k, k))}
%o A226733 for(n=0, 30, print1(a(n), ", "))
%o A226733 (PARI) {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(4*n-2*k, k))}
%o A226733 for(n=0, 30, print1(a(n), ", "))
%o A226733 (PARI) {a(n)=sum(k=0, n, binomial(4*n+2*k, n-k)*binomial(-2*k, k))}
%o A226733 for(n=0, 30, print1(a(n), ", "))
%Y A226733 Cf. A147855, A183160, A226705, A002293.
%Y A226733 Cf. A005810, A078995, A147855, A226761, A385605, A386811.
%K A226733 nonn
%O A226733 0,2
%A A226733 _Paul D. Hanna_, Jun 16 2013