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
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Magma
[Binomial(2*n+1, n+1)*Binomial(n+2, 2): n in [0..30]]; // G. C. Greubel, Feb 12 2019
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Mathematica
Table[Binomial[2n+1, n+1]Binomial[n+2, 2], {n, 0, 30}] -
PARI
a(n)=binomial(2*n+1,n+1)*binomial(n+2,2)
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Sage
[binomial(2*n+1, n+1)*binomial(n+2, 2) for n in (0..30)] # G. C. Greubel, Feb 12 2019
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)