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%I #10 Nov 05 2021 13:10:20
%S 1,1,4,16,71,336,1660,8464,44207,235306,1271807,6961307,38508659,
%T 214950425,1209170536,6848080767,39014400171,223439516338,
%U 1285660965508,7428738358924,43087099589998,250766507928988,1464026402082801
%N G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3).
%H Seiichi Manyama, <a href="/A161797/b161797.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n} C(n,k)/(n-k+1) * C(n+2*k-1,n-k).
%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then
%F a(n,m) = Sum_{k=0..n} C(n+m-1,k)*m/(n-k+m) * C(n+2*k-1,n-k).
%F G.f.: A(x) = (1/x)*serreverse[x/(1 + x/(1 - x)^3)].
%F Recurrence: 3*(n+1)*(3*n - 2)*(3*n + 2)*(2145*n^4 - 14355*n^3 + 33844*n^2 - 32668*n + 10380)*a(n) = 3*(115830*n^7 - 833085*n^6 + 2195691*n^5 - 2521863*n^4 + 998671*n^3 + 259048*n^2 - 263292*n + 41520)*a(n-1) + 3*(n-2)*(19305*n^6 - 129195*n^5 + 315651*n^4 - 367201*n^3 + 219176*n^2 - 66584*n + 7608)*a(n-2) + 3*(n-3)*(n-2)*(64350*n^5 - 334125*n^4 + 546005*n^3 - 255608*n^2 - 71320*n + 54328)*a(n-3) - 23*(n-4)*(n-3)*(n-2)*(2145*n^4 - 5775*n^3 + 3649*n^2 + 535*n - 654)*a(n-4). - _Vaclav Kotesovec_, Nov 18 2017
%F a(n) ~ sqrt(sqrt((159 + 100*sqrt(3))/13) - 2 - 5/sqrt(3)) * (3 + 2*sqrt(3) + sqrt(153 + 100*sqrt(3))/3)^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 5/2)). - _Vaclav Kotesovec_, Nov 18 2017
%t Table[Sum[Binomial[n,k]/(n-k+1) * Binomial[n+2*k-1,n-k], {k,0,n}], {n,0,30}] (* _Vaclav Kotesovec_, Nov 18 2017 *)
%o (PARI) {a(n,m=1)=sum(k=0,n,binomial(n+m-1,k)*m/(n-k+m)*binomial(n+2*k-1,n-k))}
%Y Cf. A109081, A321798, A321799.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 19 2009
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