A366977 -id:A366977 - cp's OEIS frontend

A366978 a(n) = Sum_{j=1..n} binomial(floor(n/j)+n,n+1).

1, 5, 17, 64, 220, 839, 3061, 11684, 44126, 169432, 648589, 2505411, 9670165, 37497431, 145502481, 566076182, 2204451031, 8599761208, 33581164151, 131296796355, 513812162117, 2012709456997, 7890502860027, 30958303856804, 121549519502347, 477555096290870, 1877411492125154

Offset: 1

Chai Wah Wu, Oct 30 2023

nonn

Superdiagonal of array in A366977.

Table[Sum[Binomial[j+n-1,n]Floor[n/j],{j,n}],{n,30}] (* Harvey P. Dale, Jul 19 2024 *)

from math import isqrt, comb
def A366978(n): return (-(s:=isqrt(n))**2*comb(s+n,n)+sum((q:=n//j)*((n+1)*comb(j+n-1,n)+comb(q+n,n)) for j in range(1,s+1)))//(n+1)

a(n) = Sum_{j=1..n} binomial(j+n-1,n)*floor(n/j).

a(n) ~ 4^n / sqrt(Pi*n). - Vaclav Kotesovec, Sep 19 2024

A366986 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} binomial(d+k-1,k).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 4, 3, 1, 5, 7, 7, 2, 1, 6, 11, 14, 6, 4, 1, 7, 16, 25, 16, 12, 2, 1, 8, 22, 41, 36, 31, 8, 4, 1, 9, 29, 63, 71, 71, 29, 15, 3, 1, 10, 37, 92, 127, 147, 85, 50, 13, 4, 1, 11, 46, 129, 211, 280, 211, 145, 52, 18, 2, 1, 12, 56, 175, 331, 498, 463, 371, 176, 74, 12, 6

Offset: 1

Seiichi Manyama, Oct 31 2023

			Square  array begins:
  1,  1,  1,  1,   1,   1,   1, ...
  2,  3,  4,  5,   6,   7,   8, ...
  2,  4,  7, 11,  16,  22,  29, ...
  3,  7, 14, 25,  41,  63,  92, ...
  2,  6, 16, 36,  71, 127, 211, ...
  4, 12, 31, 71, 147, 280, 498, ...
  2,  8, 29, 85, 211, 463, 925, ...

Columns k=0..5 give A000005, A000203, A007437, A059358, A073570, A101289.
T(n,n-1) gives A332508.
T(n,n) gives A343548.
Cf. A366977.

T(n, k) = sumdiv(n, d, binomial(d+k-1, k));

G.f. of column k: Sum_{j>=1} x^j/(1 - x^j)^(k+1).

cp's OEIS Frontend

A366978 a(n) = Sum_{j=1..n} binomial(floor(n/j)+n,n+1).

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