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%I #30 Apr 05 2024 13:07:30
%S 1,3,22,174,1444,12323,107104,942952,8381596,75053100,676017962,
%T 6118171326,55591175956,506805088026,4633571685968,42468065811884,
%U 390071875757852,3589637747968964,33089300640166360,305476314574338648,2823932709938708824,26137341654281261347
%N G.f.: 1 / (1 + 4*x*G(x)^2 - 7*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%H G. C. Greubel, <a href="/A147855/b147855.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n} C(k, n-k) * C(4*n-k, k).
%F a(n) = Sum_{k=0..n} C(n+k, n-k) * C(3*n-k, k).
%F a(n) = Sum_{k=0..n} C(2*n+k, n-k) * C(2*n-k, k).
%F a(n) = Sum_{k=0..n} C(3*n+k, n-k) * C(n-k, k).
%F a(n) = Sum_{k=0..n} C(4*n+k, n-k) * C(-k, k).
%F G.f.: 1 / (1 - 3*x*G(x)^2 - 7*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%F a(n) ~ 2^(8*n+5/2)/(5*sqrt(Pi*n)*3^(3*n+1/2)). - _Vaclav Kotesovec_, Jun 16 2013
%F From _Seiichi Manyama_, Apr 05 2024: (Start)
%F a(n) = Sum_{k=0..floor(n/2)} binomial(4*n-2*k-1,n-2*k).
%F a(n) = [x^n] 1/((1-x^2) * (1-x)^(3*n)). (End)
%e G.f.: A(x) = 1 + 3*x + 22*x^2 + 174*x^3 + 1444*x^4 + 12323*x^5 +...
%e A related series is G(x) = 1 + x*G(x)^4, where
%e G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%e G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 + 17710*x^6 +...
%e G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
%e such that A(x) = 1/(1 + 4*x*G(x)^2 - 7*x*G(x)^3).
%t Table[Sum[Binomial[2*n+k,n-k]*Binomial[2*n-k,k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 16 2013 *)
%o (PARI) {a(n)=sum(k=0, n, binomial(2*n+k, n-k)*binomial(2*n-k, k))}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=sum(k=0, n, binomial(k, n-k)*binomial(4*n-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^4+x*O(x^n)); polcoeff(1/(1+4*x*G^2-7*x*G^3), 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^4+x*O(x^n)); polcoeff(1/(1-3*x*G^2-7*x^2*G^6), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A226733, A226761, A226705, A002293.
%Y Cf. A026641, A183160.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 16 2013
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