cp's OEIS Frontend

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).

%I A366977 #21 Oct 31 2023 03:32:56
%S A366977 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,
%T A366977 1,9,30,64,78,73,41,20,1,10,38,93,135,149,102,56,23,1,11,47,130,220,
%U A366977 282,234,152,69,27,1,12,57,176,341,500,493,379,204,87,29
%N 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).
%F A366977 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).
%F A366977 G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j/(1 - x^j)^(k+1). - _Seiichi Manyama_, Oct 30 2023
%e A366977 Array begins:
%e A366977    1,  1,   1,   1,   1,   1,    1,    1,    1,    1, ...
%e A366977    3,  4,   5,   6,   7,   8,    9,   10,   11,   12, ...
%e A366977    5,  8,  12,  17,  23,  30,   38,   47,   57,   68, ...
%e A366977    8, 15,  26,  42,  64,  93,  130,  176,  232,  299, ...
%e A366977   10, 21,  42,  78, 135, 220,  341,  507,  728, 1015, ...
%e A366977   14, 33,  73, 149, 282, 500,  839, 1344, 2070, 3083, ...
%e A366977   16, 41, 102, 234, 493, 963, 1764, 3061, 5074, 8089, ...
%o A366977 (Python)
%o A366977 from math import isqrt, comb
%o A366977 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)
%o A366977 def A366977_gen(): # generator of terms
%o A366977     return (A366977_T(k+1, n-k-1) for n in count(1) for k in range(n))
%o A366977 A366977_list = list(islice(A366977_gen(),30))
%Y A366977 First superdiagonal is A366978.
%Y A366977 Columns k=0..4 give A006218, A024916, A364970, A365409, A365439.
%K A366977 nonn,tabl
%O A366977 1,3
%A A366977 _Chai Wah Wu_, Oct 30 2023