A366977 Array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} binomial(floor(n/j)+k,k+1).

1, 1, 3, 1, 4, 5, 1, 5, 8, 8, 1, 6, 12, 15, 10, 1, 7, 17, 26, 21, 14, 1, 8, 23, 42, 42, 33, 16, 1, 9, 30, 64, 78, 73, 41, 20, 1, 10, 38, 93, 135, 149, 102, 56, 23, 1, 11, 47, 130, 220, 282, 234, 152, 69, 27, 1, 12, 57, 176, 341, 500, 493, 379, 204, 87, 29

Offset

1,3

Links

Table of n, a(n) for n=1..66.

Formula

T(n,k) = Sum_{j=1..n} binomial(j+k-1,k)*floor(n/j) = (Sum_{j=1..floor(sqrt(n))} [floor(n/j)*((k+1)*binomial(j+k-1,k)+binomial(floor(n/j)+k,k))] - floor(sqrt(n))^2*binomial(floor(sqrt(n))+k,k))/(k+1).

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j/(1 - x^j)^(k+1). - Seiichi Manyama, Oct 30 2023

Example

Array begins:
   1,  1,   1,   1,   1,   1,    1,    1,    1,    1, ...
   3,  4,   5,   6,   7,   8,    9,   10,   11,   12, ...
   5,  8,  12,  17,  23,  30,   38,   47,   57,   68, ...
   8, 15,  26,  42,  64,  93,  130,  176,  232,  299, ...
  10, 21,  42,  78, 135, 220,  341,  507,  728, 1015, ...
  14, 33,  73, 149, 282, 500,  839, 1344, 2070, 3083, ...
  16, 41, 102, 234, 493, 963, 1764, 3061, 5074, 8089, ...

Prog

(Python)
from math import isqrt, comb
def A366977_T(n, k): return (-(s:=isqrt(n))**2*comb(s+k, k)+sum((q:=n//j)*((k+1)*comb(j+k-1, k)+comb(q+k, k)) for j in range(1, s+1)))//(k+1)
def A366977_gen(): # generator of terms
    return (A366977_T(k+1, n-k-1) for n in count(1) for k in range(n))
A366977_list = list(islice(A366977_gen(), 30))

Crossrefs

First superdiagonal is A366978.

Columns k=0..4 give A006218, A024916, A364970, A365409, A365439.

Sequence in context: A298890 A016473 A343516 * A029637 A097207 A266101

Adjacent sequences: A366974 A366975 A366976 * A366978 A366979 A366980

Keyword

nonn,tabl

Author

Chai Wah Wu, Oct 30 2023

Status

approved