A087413 -id:A087413 - cp's OEIS frontend

A188675 Partial sums of the binomial coefficients binomial(3*n,n) (A005809).

1, 4, 19, 103, 598, 3601, 22165, 138445, 873916, 5560741, 35605756, 229142476, 1480820176, 9603245620, 62463474700, 407330900284, 2662179813931, 17433248900656, 114359597479261, 751343566800961, 4943188072606456

Offset: 0

Emanuele Munarini, Apr 08 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000(terms 1..100 from Vincenzo Librandi)

Cf. A001764, A005809, A104859, A188676, A188678 - A188687.
Cf. A263134: Sum_{k=0..n} binomial(3*k+1,k).
Cf. A087413: Sum_{k=0..n} binomial(3*k+2,k).

Table[Sum[Binomial[3k, k], {k, 0, n}], {n, 0, 20}]
Accumulate[Table[Binomial[3n,n],{n,0,20}]] (* Nearly 300 times faster than the program above. *) (* Harvey P. Dale, Sep 14 2024 *)

makelist(sum(binomial(3*k,k),k,0,n),n,0,20);

for(n=0,25, print1(sum(k=0,n, binomial(3*k,k)), ", ")) \\ G. C. Greubel, Jan 27 2017

a(n) = Sum_{k=0..n} binomial(3*k,k).
Recurrence: 2*(n+2)*(2n+3)*a(n+2)-(31*n^2+95*n+72)*a(n+1)+3*(3*n+4)(3*n+5)*a(n)=0.
G.f.: 2*cos((1/3)*arcsin(3*sqrt(3*x)/2))/((1-x)*sqrt(4-27*x)).
a(n) ~ sqrt(3)*27^(n+1)/(46*4^n*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012

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

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Oct 10 2015

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

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

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

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).

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

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

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

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

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

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

cp's OEIS Frontend

A188675 Partial sums of the binomial coefficients binomial(3*n,n) (A005809).

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