A128321 Column 0 of triangle A128320.

1, 1, 4, 17, 98, 622, 4512, 35373, 300974, 2722070, 26118056, 263266346, 2780054884, 30586452652, 349724463584, 4141218303165, 50678688359190, 639387728054310, 8302396672724280, 110754894628585950

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..600

Formula

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

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)).

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

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

Mathematica

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)];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jun 25 2024 *)

Prog

(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))}
(Magma)
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
(SageMath)
@CachedFunction
def a(n): # a = A128321
    if n<3: return (1, 1, 4)[n]
    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)
[a(n) for n in range(31)] # G. C. Greubel, Jun 25 2024

Crossrefs

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

Cf. A000108 (Catalan numbers).

Sequence in context: A249078 A353546 A024052 * A290352 A091635 A306160

Adjacent sequences: A128318 A128319 A128320 * A128322 A128323 A128324

Keyword

nonn

Author

Paul D. Hanna, Feb 25 2007

Status

approved