A051033 -id:A051033 - cp's OEIS frontend

A051036 a(n) = binomial(n, floor(n/4)).

1, 1, 1, 1, 4, 5, 6, 7, 28, 36, 45, 55, 220, 286, 364, 455, 1820, 2380, 3060, 3876, 15504, 20349, 26334, 33649, 134596, 177100, 230230, 296010, 1184040, 1560780, 2035800, 2629575, 10518300, 13884156, 18156204, 23535820, 94143280, 124403620

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001405, A051033, A051052, A051053, A062947.

a[n_] := Binomial[n, Floor[n/4]]; Array[a, 50, 0] (* Enrique Pérez Herrero, Mar 06 2012 *)

from math import comb
def A051036(n): return comb(n,n>>2) # Chai Wah Wu, Feb 02 2023

A051052 a(n) = binomial(n, floor(n/5)).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 6, 7, 8, 9, 45, 55, 66, 78, 91, 455, 560, 680, 816, 969, 4845, 5985, 7315, 8855, 10626, 53130, 65780, 80730, 98280, 118755, 593775, 736281, 906192, 1107568, 1344904, 6724520, 8347680, 10295472, 12620256, 15380937, 76904685

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001405, A051033, A051036, A051053, A062947.

Table[Binomial[n,Floor[n/5]],{n,0,40}] (* Harvey P. Dale, Aug 06 2017 *)

A051053 a(n) = binomial(n, floor(n/6)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 7, 8, 9, 10, 11, 66, 78, 91, 105, 120, 136, 816, 969, 1140, 1330, 1540, 1771, 10626, 12650, 14950, 17550, 20475, 23751, 142506, 169911, 201376, 237336, 278256, 324632, 1947792, 2324784, 2760681, 3262623, 3838380, 4496388

Offset: 0

N. J. A. Sloane

nonn

Robert Israel, Table of n, a(n) for n = 0..5117

Cf. A001405, A051033, A051036, A051052.

seq(binomial(n, floor(n/6)), n=0..60); # Robert Israel, Mar 11 2018

Table[Binomial[n,Floor[n/6]],{n,0,50}] (* Harvey P. Dale, Dec 18 2013 *)

From Robert Israel, Mar 11 2018: (Start)
Let n = 6*k+j, 0 <= j <= 5.
a(n+6)*(k+1)*Product_{m=1..5} (5*k+j+m) = a(n)*Product_{m=1..6} (6*k+j+m).
a(n) ~ sqrt(3/(5*Pi*k))*(6/5)^j*(6^6/5^5)^k as k -> infinity. (End)

A047193 Number of nonempty subsets of {1,2,...,n} in which exactly 1/2 of the elements are <= n/3.

Original entry on oeis.org

0, 0, 2, 3, 4, 14, 20, 27, 83, 119, 164, 494, 714, 1000, 3002, 4367, 6187, 18563, 27131, 38759, 116279, 170543, 245156, 735470, 1081574, 1562274, 4686824, 6906899, 10015004, 30045014, 44352164, 64512239, 193536719, 286097759

Offset: 1

Clark Kimberling

nonn

a(n) = C(n, [n/3]) - 1 = A051033(n) - 1. - Ralf Stephan, Mar 16 2004

A066704 Triangle with a(n,k) = C(n,floor(n/k)) with n>=k>=1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 6, 4, 4, 1, 10, 5, 5, 5, 1, 20, 15, 6, 6, 6, 1, 35, 21, 7, 7, 7, 7, 1, 70, 28, 28, 8, 8, 8, 8, 1, 126, 84, 36, 9, 9, 9, 9, 9, 1, 252, 120, 45, 45, 10, 10, 10, 10, 10, 1, 462, 165, 55, 55, 11, 11, 11, 11, 11, 11, 1, 924, 495, 220, 66, 66, 12, 12, 12, 12, 12, 12

Offset: 1

Henry Bottomley, Jan 14 2002

			Rows start:
  1;
  1,  2;
  1,  3, 3;
  1,  6, 4, 4;
  1, 10, 5, 5, 5;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Row sums are A051054.
Columns include (most of) A000012, A001405, A051033, A051036, A051052, A051053, A062947 etc.
n appears A008619 times in the n-th row.
Cf. A060539.

A355703 a(n) = binomial(n, floor(log(n))).

Original entry on oeis.org

1, 1, 3, 4, 5, 6, 7, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180, 16215, 17296, 18424, 19600, 20825

Offset: 1

Christoph B. Kassir, Jul 14 2022

nonn

Mathematics Stack Exchange, How to compute asymptotic growth of binomial(n, log(n)).

Cf. A000195, A007318.

Cf. A001405, A051033, A051036, A051052, A051053.

a:= n-> binomial(n, ilog(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 31 2022

a[n_] := Binomial[n, Floor[Log[n]]]; Array[a, 50] (* Amiram Eldar, Jul 31 2022 *)

a(n) = binomial(n, floor(log(n))); \\ Michel Marcus, Jul 31 2022

from numpy import log
from math import comb, floor
for n in range(1, 50):
    x = comb(n, floor(log(n)))
    print("{}, ".format(x), end='')

cp's OEIS Frontend

A051036 a(n) = binomial(n, floor(n/4)).

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A051052 a(n) = binomial(n, floor(n/5)).

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A051053 a(n) = binomial(n, floor(n/6)).

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A047193 Number of nonempty subsets of {1,2,...,n} in which exactly 1/2 of the elements are <= n/3.

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Formula

A066704 Triangle with a(n,k) = C(n,floor(n/k)) with n>=k>=1.

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A355703 a(n) = binomial(n, floor(log(n))).

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Author

Keywords

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Maple

Mathematica

PARI

Python