A085374 a(n) = binomial(2n+1, n+1)*binomial(n+3, 3).

1, 12, 100, 700, 4410, 25872, 144144, 772200, 4011150, 20323160, 100876776, 492156392, 2366136500, 11232648000, 52739956800, 245240799120, 1130632213590, 5172827121000, 23504600427000, 106141827191400, 476627347816620, 2129348151284640, 9468445336740000

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Jun 26 2003

Keywords

• easy
• nonn
• changed

Crossrefs

Cf. A001700, A002457, A085373.

Cf. A000292, A000984.

Programs

Maple

seq(binomial(n+2,3)/2*binomial(2*n,n), n=1..20); # Zerinvary Lajos, Jan 18 2007

Mathematica

Table[Binomial[2 n + 1, n + 1]Binomial[n + 3, 3], {n, 0, 30}]

Formula

a(n) = A000292(n+1)*A000984(n+1)/2. - Zerinvary Lajos, Jan 18 2007, corrected Aug 09 2015

D-finite with recurrence n*(n+1)*a(n) - 2*(n+3)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Aug 09 2015

G.f.: 2F1(3/2,4;2;4x). - R. J. Mathar, Aug 09 2015

a(n) ~ 2^(2*n)*n^(5/2)/(3*sqrt(Pi)). - Stefano Spezia, Aug 31 2025