A225615 -id:A225615 - cp's OEIS frontend

A225612 Partial sums of the binomial coefficients C(4*n,n).

1, 5, 33, 253, 2073, 17577, 152173, 1336213, 11854513, 105997793, 953658321, 8622997453, 78291531921, 713305091521, 6518037055321, 59712126248041, 548239063327621, 5043390644753269, 46475480410336709, 428936432074181109, 3964252574286355429

Offset: 0

Vaclav Kotesovec, Aug 06 2013

Generally (for p>1), partial sums of the binomial coefficients C(p*n,n) are asymptotic to (1/(1-(p-1)^(p-1)/p^p)) * sqrt(p/(2*Pi*n*(p-1))) * (p^p/(p-1)^(p-1))^n.

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

Cf. A006134 (p=2), A188675 (p=3), A225615 (p=5).

A225612:=n->add(binomial(4*k,k), k=0..n): seq(A225612(n), n=0..30); # Wesley Ivan Hurt, Apr 01 2017

Table[Sum[Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]
Accumulate[Table[Binomial[4n,n],{n,0,20}]] (* Harvey P. Dale, Feb 01 2015 *)

for(n=0,50, print1(sum(k=0,n, binomial(4*k,k)), ", ")) \\ G. C. Greubel, Apr 01 2017

Recurrence: 3*n*(3*n-2)*(3*n-1)*a(n) = (283*n^3 - 411*n^2 + 182*n - 24)*a(n-1) - 8*(2*n-1)*(4*n-3)*(4*n-1)*a(n-2).

a(n) ~ 2^(8*n+17/2)/(229*sqrt(Pi*n)*3^(3*n+1/2)).

A358146 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 4, 1, 1, 5, 19, 29, 5, 1, 1, 6, 33, 103, 99, 6, 1, 1, 7, 51, 253, 598, 351, 7, 1, 1, 8, 73, 506, 2073, 3601, 1275, 8, 1, 1, 9, 99, 889, 5351, 17577, 22165, 4707, 9, 1, 1, 10, 129, 1429, 11515, 58481, 152173, 138445, 17577, 10, 1

Offset: 0

Seiichi Manyama, Oct 31 2022

			Square array begins:
  1, 1,   1,    1,     1,     1, ...
  1, 2,   3,    4,     5,     6, ...
  1, 3,   9,   19,    33,    51, ...
  1, 4,  29,  103,   253,   506, ...
  1, 5,  99,  598,  2073,  5351, ...
  1, 6, 351, 3601, 17577, 58481, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A000012, A001477(n+1), A006134, A188675, A225612, A225615.
Main diagonal gives A226391.
Cf. A358050.

T(n, k) = sum(j=0, n, binomial(k*j, j));

cp's OEIS Frontend

A225612 Partial sums of the binomial coefficients C(4*n,n).

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