A051054 -id:A051054 - cp's OEIS frontend

A273160 a(n) = Sum_{k=1..n} C(n, floor((n-k)/k)).

0, 1, 3, 5, 10, 13, 30, 39, 84, 143, 290, 424, 1120, 1697, 3521, 6633, 14012, 22721, 53529, 88607, 189749, 356024, 730697, 1256272, 2889484, 5006918, 10399649, 19706342, 40954970, 72663140, 160499071, 286286349, 597385864, 1134282375, 2340807724, 4251969339

Offset: 0

Wesley Ivan Hurt, May 16 2016

Cf. A051054, A273161.

A273160:=n->add(binomial(n,floor((n-i)/i)), i=1..n): seq(A273160(n), n=0..50);

Table[Sum[Binomial[n, Floor[(n - i)/i]], {i, n}], {n, 0, 40}]

a(n) ~ 2^(n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 28 2021

A273161 a(n) = Sum_{k=1..n} C(n-k, floor((n-k)/k)).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 13, 19, 34, 61, 108, 174, 384, 641, 1166, 2337, 4458, 7828, 16421, 29346, 57231, 114126, 215915, 396491, 839324, 1549146, 2983185, 5978656, 11628952, 21812113, 45099914, 84842925, 166417181, 332267593, 647614074, 1234586894, 2538571022

Offset: 0

Wesley Ivan Hurt, May 16 2016

Cf. A051054, A273160.

A273161:=n->add(binomial(n-i,floor((n-i)/i)), i=1..n): seq(A273161(n), n=0..50);

Table[Sum[Binomial[n - i, Floor[(n - i)/i]], {i, n}], {n, 0, 40}]

a(n) ~ 2^(n - 3/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 28 2021

A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

Original entry on oeis.org

1, 1, 2, 9, 96, 1250, 64800, 1764735, 224788480, 22499086176, 6123600000000, 408514437465750, 1308805762115174400, 133962125607455951520, 99335199198879310098432, 113040832521732593994140625, 425230288403106927476736000000, 72623663171934137824096600064000

Offset: 0

Vaclav Kotesovec, Jun 20 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..160

Cf. A010786, A034841, A051054, A092143, A272093.

[n eq 0 select 1 else (&*[Binomial(n,Floor(n/j)): j in [1..n]]): n in [0..30]]; // G. C. Greubel, Feb 05 2024

Table[Product[Binomial[n, Floor[n/k]], {k, 1, n}], {n, 0, 20}]
Table[Product[((n + 1)/k - 1)^Floor[n/k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 24 2021 *)

[product(binomial(n,(n//j)) for j in range(1,n+1)) for n in range(31)] # G. C. Greubel, Feb 05 2024

log(a(n)) ~ n * log(n)^2 / 2. - Vaclav Kotesovec, Jun 21 2021

a(n) = Product_{k=1..n} ((n+1)/k - 1)^floor(n/k). - Vaclav Kotesovec, Jun 24 2021

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.

cp's OEIS Frontend

A273160 a(n) = Sum_{k=1..n} C(n, floor((n-k)/k)).

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Maple

Mathematica

Formula

A273161 a(n) = Sum_{k=1..n} C(n-k, floor((n-k)/k)).

Views

Author

Keywords

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Programs

Maple

Mathematica

Formula

A345466 a(n) = Product_{k=1..n} binomial(n, floor(n/k)).

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Magma

Mathematica

SageMath

Formula

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

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