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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).
2
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
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
KEYWORD
nonn,tabl
AUTHOR
Chai Wah Wu, Oct 30 2023
STATUS
approved