A183161 Self-convolution equals A183160.
1, 1, 5, 26, 145, 841, 5006, 30350, 186537, 1158685, 7258145, 45779420, 290399030, 1851032314, 11847434810, 76100034106, 490343021881, 3168174174105, 20520045125681, 133197288251330, 866293102078525, 5644234561103785
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 145*x^4 + 841*x^5 + 5006*x^6 +... 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 +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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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)
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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
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PARI
A002426(n)=sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k)) {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 for(n=0,30,print1(a(n),", "))
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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) for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2013
Formula
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).
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
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
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
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
From Vaclav Kotesovec, Oct 05 2020: (Start)
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).
a(n) ~ 3^(3*n + 3/4) / (Gamma(1/4) * n^(3/4) * 2^(2*n + 3/2)). (End)
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