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A226705 G.f.: 1 / (1 + 12*x*G(x)^4 - 16*x*G^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

%I A226705 #35 Aug 15 2025 15:58:02
%S A226705 1,4,48,600,7856,105684,1447392,20075416,281086416,3964453368,
%T A226705 56240518128,801624722232,11470976280960,164691196943212,
%U A226705 2371222443727584,34224696393237360,495036708728067088,7173892793100898728,104135761805147016096,1513892435551302963792
%N A226705 G.f.: 1 / (1 + 12*x*G(x)^4 - 16*x*G^5) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
%H A226705 Vincenzo Librandi, <a href="/A226705/b226705.txt">Table of n, a(n) for n = 0..200</a>
%F A226705 a(n) = Sum_{k=0..n} C(2*k, n-k) * C(6*n-2*k, k).
%F A226705 a(n) = Sum_{k=0..n} C(n+2*k, n-k) * C(5*n-2*k, k).
%F A226705 a(n) = Sum_{k=0..n} C(2*n+2*k, n-k) * C(4*n-2*k, k).
%F A226705 a(n) = Sum_{k=0..n} C(3*n+2*k, n-k) * C(3*n-2*k, k).
%F A226705 a(n) = Sum_{k=0..n} C(4*n+2*k, n-k) * C(2*n-2*k, k).
%F A226705 a(n) = Sum_{k=0..n} C(5*n+2*k, n-k) * C(n-2*k, k).
%F A226705 a(n) = Sum_{k=0..n} C(6*n+2*k, n-k) * C(-2*k, k).
%F A226705 Self-convolution of A226706.
%F A226705 G.f.: 1 / (1 - 4*x*G(x)^4 - 16*x^2*G(x)^10) where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
%F A226705 a(n) ~ 2^(6*n-2)*3^(6*n+3/2)/(5^(5*n+1/2)*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jun 16 2013
%F A226705 From _Seiichi Manyama_, Aug 05 2025: (Start)
%F A226705 a(n) = [x^n] 1/((1+2*x) * (1-x)^(5*n+1)).
%F A226705 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(6*n+1,k).
%F A226705 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(5*n+k,k). (End)
%F A226705 From _Seiichi Manyama_, Aug 14 2025: (Start)
%F A226705 a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(6*n+1,k) * binomial(6*n-k,n-k).
%F A226705 G.f.: G(x)^2/((-2+3*G(x)) * (6-5*G(x))) where G(x) = 1+x*G(x)^6 is the g.f. of A002295. (End)
%F A226705 G.f.: B(x)^2/(1 + 4*(B(x)-1)/3), where B(x) is the g.f. of A004355. - _Seiichi Manyama_, Aug 15 2025
%e A226705 G.f.: A(x) = 1 + 4*x + 48*x^2 + 600*x^3 + 7856*x^4 + 105684*x^5 +...
%e A226705 A related series is G(x) = 1 + x*G(x)^6, where
%e A226705 G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
%e A226705 G(x)^4 = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...
%e A226705 G(x)^5 = 1 + 5*x + 40*x^2 + 385*x^3 + 4095*x^4 + 46376*x^5 +...
%e A226705 such that A(x) = 1/(1 + 12*x*G(x)^4 - 16*x*G^5).
%t A226705 Table[Sum[Binomial[3*n+2*k,n-k]*Binomial[3*n-2*k,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 16 2013 *)
%o A226705 (PARI) {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^6+x*O(x^n)); polcoeff(1/(1+12*x*G^4-16*x*G^5), n)}
%o A226705 for(n=0, 30, print1(a(n), ", "))
%o A226705 (PARI) {a(n)=local(G=1+x); for(i=0, n,G=1+x*G^6+x*O(x^n)); polcoeff(1/(1-4*x*G^4-16*x^2*G^10), n)}
%o A226705 for(n=0, 30, print1(a(n), ", "))
%o A226705 (PARI) {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(6*n-2*k, k))}
%o A226705 for(n=0, 30, print1(a(n), ", "))
%o A226705 (PARI) {a(n)=sum(k=0, n, binomial(3*n +2*k, n-k)*binomial(3*n-2*k, k))}
%o A226705 for(n=0, 30, print1(a(n), ", "))
%o A226705 (PARI) {a(n)=sum(k=0, n, binomial(6*n +2*k, n-k)*binomial(-2*k, k))}
%o A226705 for(n=0, 30, print1(a(n), ", "))
%Y A226705 Cf. A226706, A183160, A147855, A002295.
%Y A226705 Cf. A226733, A226751.
%Y A226705 Cf. A004355, A079679.
%K A226705 nonn
%O A226705 0,2
%A A226705 _Paul D. Hanna_, Jun 15 2013