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Column 0 of triangle A128320.
5

%I #14 Jun 25 2024 19:24:32

%S 1,1,4,17,98,622,4512,35373,300974,2722070,26118056,263266346,

%T 2780054884,30586452652,349724463584,4141218303165,50678688359190,

%U 639387728054310,8302396672724280,110754894628585950

%N Column 0 of triangle A128320.

%H G. C. Greubel, <a href="/A128321/b128321.txt">Table of n, a(n) for n = 0..600</a>

%F a(n) = Sum_{k=0..floor(n/2)} A000108(n-k)*A000108(k)*(k+1)!*C(n,2*k).

%F a(n) = Sum_{k=0..floor((n+1)/2)} ((k+1)!*C(2*(n-k), n-k)*C(2*k, k)*C(n, 2*k))/((k+1)*(n-k+1)).

%F a(n) = ( -(n-1)*(n-2)*a(n-1) + 4*(3*n^2 -5*n +1)*a(n-2) + 8*n*(n-2)^2* a(n-3) )/(n+1), with a(0) = 1, a(1) = 1, a(2) = 4. - _G. C. Greubel_, Jun 25 2024

%F a(n) ~ 2^(3*n/2 + 1) * exp(sqrt(2*n) - n/2 - 1/2) * n^((n-3)/2) / sqrt(Pi) * (1 - 7/(3*sqrt(2*n))). - _Vaclav Kotesovec_, Jun 25 2024

%t a[n_]:= a[n]= If[n<3, (n!)^2, (-(n-1)*(n-2)*a[n-1] +4*(3*n^2-5*n +1)*a[n-2] + 8*(n-2)^2*n*a[n-3])/(n+1)];

%t Table[a[n], {n,0,40}] (* _G. C. Greubel_, Jun 25 2024 *)

%o (PARI) {a(n)=sum(k=0,n\2,binomial(2*n-2*k,n-k)/(n-k+1)*binomial(2*k,k)/(k+1) *(k+1)!*binomial(n,2*k))}

%o (Magma)

%o I:=[1,1,4]; [n le 3 select I[n] else (-(n-2)*(n-3)*Self(n-1) + 4*(3*(n-2)^2+n-3)*Self(n-2) + 8*(n-3)^2*(n-1)*Self(n-3))/n: n in [1..30]]; // _G. C. Greubel_, Jun 25 2024

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A128321

%o if n<3: return (1,1,4)[n]

%o else: return (-(n-1)*(n-2)*a(n-1) + 4*(3*n^2-5*n+1)*a(n-2) + 8*(n-2)^2*n*a(n-3))/(n+1)

%o [a(n) for n in range(31)] # _G. C. Greubel_, Jun 25 2024

%Y Cf. A128320 (triangle), A128322 (column 1), A128323 (column 2), A128324 (row sums); variant: A115081.

%Y Cf. A000108 (Catalan numbers).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 25 2007