A263134 a(n) = Sum_{k=0..n} binomial(3*k+1,k).

1, 5, 26, 146, 861, 5229, 32361, 202905, 1284480, 8191380, 52543545, 338641305, 2191124301, 14224347181, 92603307541, 604342068085, 3952451061076, 25898039418496, 169977746765071, 1117287239602471, 7353933943361866, 48461930821297546

Offset: 0

Author

Bruno Berselli, Oct 10 2015

Keywords

• nonn
• easy

Comments

Primes in sequence: 5, 92603307541, 52176309488123582020412161, ...

a(n) is divisible by n for n = 1, 2, 8, 55, 82, 171, 210, 1060, 1141, ...

Links

• Bruno Berselli and G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 form Bruno Berselli)

Crossrefs

Partial sums of A045721.

Cf. A079309: Sum_{k=0..n} binomial(2*k+1,k).

Cf. A188675: Sum_{k=0..n} binomial(3*k,k).

Cf. A087413: Sum_{k=0..n} binomial(3*k+2,k).

Programs

Magma

[&+[Binomial(3*k+1,k): k in [0..n]]: n in [0..25]];

Mathematica

Table[Sum[Binomial[3 k + 1, k], {k, 0, n}], {n, 0, 25}]

Maxima

makelist(sum(binomial(3*k+1,k),k,0,n),n,0,25);

PARI

a(n) = sum(k=0, n, binomial(3*k+1,k)) \\ Colin Barker, Oct 16 2015

Sage

[sum(binomial(3*k+1,k) for k in (0..n)) for n in (0..25)]

Formula

Recurrence: 2*n*(2*n + 1)*a(n) = (31*n^2 + 2*n - 3)*a(n-1) - 3*(3*n - 1)*(3*n + 1)*a(n-2). - Vaclav Kotesovec, Oct 11 2015

a(n) ~ 27^(n + 3/2)/(23*sqrt(Pi*n)*4^(n + 1)). - Vaclav Kotesovec, Oct 11 2015