A356124 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} j^k * binomial(floor(n/j)+1,2).
1, 1, 4, 1, 5, 8, 1, 7, 11, 15, 1, 11, 19, 23, 21, 1, 19, 41, 47, 33, 33, 1, 35, 103, 125, 77, 57, 41, 1, 67, 281, 395, 255, 149, 71, 56, 1, 131, 799, 1373, 1025, 555, 205, 103, 69, 1, 259, 2321, 5027, 4503, 2537, 905, 325, 130, 87, 1, 515, 6823, 18965, 20657, 12867, 4945, 1585, 442, 170, 99
Offset: 1
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 4, 5, 7, 11, 19, 35, 67, ... 8, 11, 19, 41, 103, 281, 799, ... 15, 23, 47, 125, 395, 1373, 5027, ... 21, 33, 77, 255, 1025, 4503, 20657, ... 33, 57, 149, 555, 2537, 12867, 68969, ...
Crossrefs
Programs
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Mathematica
T[n_, k_] := Sum[j^k * Binomial[Floor[n/j] + 1, 2], {j, 1, n}]; Table[T[k, n - k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 28 2022 *) -
PARI
T(n, k) = sum(j=1, n, j^k*binomial(n\j+1, 2));
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PARI
T(n, k) = sum(j=1, n, j*sigma(j, k-1));
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Python
from itertools import count, islice from math import isqrt from sympy import bernoulli def A356124_T(n,k): return ((s:=isqrt(n))*(s+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)*(q+1))+(w*(bernoulli(k+1,q+1)-bernoulli(k+1))<<1) for w in range(1,s+1)))//(k+1)>>1 def A356124_gen(): # generator of terms return (A356124_T(k+1,n-k-1) for n in count(1) for k in range(n)) A356124_list = list(islice(A356124_gen(),30)) # Chai Wah Wu, Oct 24 2023
Formula
G.f. of column k: (1/(1-x)) * Sum_{j>=1} j^k * x^j/(1 - x^j)^2.
T(n,k) = Sum_{j=1..n} j * sigma_{k-1}(j).