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

0,2

Links

Table of n, a(n) for n=0..22.

Formula

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

From R. J. Mathar, Aug 09 2015: (Start)

D-finite with recurrence n*(n+1)*a(n) - 2*(n+3)*(2*n+1)*a(n-1) = 0.

G.f.: 2F1(3/2,4;2;4x). (End)

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

From Amiram Eldar, Sep 06 2025: (Start)

a(n) = A001700(n) * A000292(n+1).

Sum_{n>=0} 1/a(n) = 10*sqrt(3)*Pi - 8*Pi^2/3 - 27.

Sum_{n>=0} (-1)^n/a(n) = 84*sqrt(5)*log(phi) - 192*log(phi)^2 - 45, where phi is the golden ratio (A001622). (End)

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}]

Crossrefs

Cf. A001700, A002457, A085373.

Cf. A000292, A000984, A001622.

Sequence in context: A367607 A123902 A358302 * A387338 A387272 A016283

Adjacent sequences: A085371 A085372 A085373 * A085375 A085376 A085377

Keyword

easy,nonn

Author

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

Status

approved