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%I #27 Oct 05 2020 04:44:40
%S 1,1,5,26,145,841,5006,30350,186537,1158685,7258145,45779420,
%T 290399030,1851032314,11847434810,76100034106,490343021881,
%U 3168174174105,20520045125681,133197288251330,866293102078525,5644234561103785
%N Self-convolution equals A183160.
%C Conjecture: a(n) is never congruent to 3 modulo 4; see A218622. - _Paul D. Hanna_, Nov 03 2012
%H Paul D. Hanna, <a href="/A183161/b183161.txt">Table of n, a(n) for n = 0..300</a>
%F Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} C(n+k,n-k)*C(2*n-k,k) = A183160(n).
%F G.f.: A(x) = 1/sqrt(1 - 2*x*G(x)^2 - 3*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - _Paul D. Hanna_, Nov 03 2012
%F G.f.: A(x) = Sum_{n>=0} A002426(n) * x^n * G(x)^(2*n), where A002426 are the central trinomial coefficients and G(x) = 1 + x*G(x)^3 = g.f. of A001764. - _Paul D. Hanna_, Nov 03 2012
%F a(n) = Sum_{k=0..n} A002426(k) * C(3*n-k,n-k) * 2*k/(3*n-k) for n>0, where A002426 are the central trinomial coefficients: A002426(n) = Sum_{k=0..[n/2]} C(n,2*k)*C(2*k,k). - _Paul D. Hanna_, Nov 04 2012
%F G.f.: A(x) = 1/sqrt(1 + 3*x*G(x) - 5*x*G(x)^2), where G(x) = 1 + x*G(x)^3 = g.f. of A001764. - _Paul D. Hanna_, Jun 16 2013
%F From _Vaclav Kotesovec_, Oct 05 2020: (Start)
%F Recurrence: 32*(n-1)*n*(2*n - 1)*(294*n^2 - 1253*n + 1310)*a(n) = 4*(n-1)*(62328*n^4 - 391468*n^3 + 870242*n^2 - 806673*n + 264534)*a(n-1) - 6*(132300*n^5 - 1169784*n^4 + 4019115*n^3 - 6676447*n^2 + 5328996*n - 1620540)*a(n-2) - 81*(n-2)*(3*n - 8)*(3*n - 7)*(294*n^2 - 665*n + 351)*a(n-3).
%F a(n) ~ 3^(3*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(2*n + 3/2)). (End)
%e G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +...
%e A(x)^2 = 1 + 2*x + 11*x^2 + 62*x^3 + 367*x^4 + 2232*x^5 + 13820*x^6 + 86662*x^7 +...+ A183160(n)*x^n +...
%o (PARI) a(n)=local(A2=sum(m=0,n,sum(k=0,m,binomial(m+k,m-k)*binomial(2*m-k,k))*x^m+x*O(x^n)));polcoeff(A2^(1/2),n)
%o (PARI) a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/sqrt(1-2*x*G^2-3*x^2*G^4), n) \\ Paul D. Hanna, Nov 03 2012
%o (PARI) A002426(n)=sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k))
%o {a(n)=if(n==0,1,sum(k=0,n,A002426(k)*binomial(3*n-k,n-k)*2*k/(3*n-k)))} \\ _Paul D. Hanna_, Nov 04 2012
%o for(n=0,30,print1(a(n),", "))
%o (PARI) a(n)=local(G=1); for(i=0, n, G=1+x*G^3+O(x^(n+1))); polcoeff(1/sqrt(1+3*x*G-5*x*G^2), n)
%o for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jun 16 2013
%Y Cf. A183160, A218622, A001764, A002426, A219197.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 27 2010
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