A085373 a(n) = binomial(2n+1, n+1)*binomial(n+2, 2).

1, 9, 60, 350, 1890, 9702, 48048, 231660, 1093950, 5080790, 23279256, 105462084, 473227300, 2106121500, 9307051200, 40873466520, 178520875830, 775924068150, 3357800061000, 14473885526100, 62168784497820, 266168518910580, 1136213440408800, 4837140552465000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

From David Callan, Nov 20 2003: (Start)

a(n-1) = Sum_{1<=i1<=i2<=...<=in<=n} (i1 + i2 + ... + in).

G.f.: (1 - x)/(1 - 4*x)^(5/2). (End)

a(n) = A119578(n+1)/2. - Zerinvary Lajos, Jun 19 2008

From Amiram Eldar, May 14 2022: (Start)

Sum_{n>=0} 1/a(n) = 4*Pi^2/9 - 4*Pi/sqrt(3) + 4.

Sum_{n>=0} (-1)^n/a(n) = 8*sqrt(5)*log(phi) - 16*log(phi)^2 - 4, where phi = A001622 is the golden ratio. (End)

a(n) ~ 4^n * n^(3/2) / sqrt(Pi). - Amiram Eldar, Sep 11 2025

D-finite with recurrence: (6 + 4*n)*a(n) + (-14 - 5*n)*a(n + 1) + (n + 2)*a(n + 2) = 0. - Robert Israel, Jun 01 2026

Mathematica

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

Prog

(PARI) a(n)=binomial(2*n+1, n+1)*binomial(n+2, 2)
(Magma) [Binomial(2*n+1, n+1)*Binomial(n+2, 2): n in [0..30]]; // G. C. Greubel, Feb 12 2019
(SageMath) [binomial(2*n+1, n+1)*binomial(n+2, 2) for n in (0..30)] # G. C. Greubel, Feb 12 2019

Crossrefs

Cf. A000984, A002544, A119578.

Sequence in context: A074431 A268965 A081904 * A241976 A082150 A385563

Adjacent sequences: A085370 A085371 A085372 * A085374 A085375 A085376

Keyword

easy,nonn

Author

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

Status

approved