A308689 -id:A308689 - cp's OEIS frontend

A308690 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*n/d - k + 1), read by antidiagonals.

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 9, 6, 1, 3, 4, 13, 6, 12, 1, 3, 4, 21, 6, 24, 8, 1, 3, 4, 37, 6, 66, 8, 15, 1, 3, 4, 69, 6, 216, 8, 41, 13, 1, 3, 4, 133, 6, 762, 8, 201, 37, 18, 1, 3, 4, 261, 6, 2784, 8, 1289, 253, 68, 12, 1, 3, 4, 517, 6, 10386, 8, 9225, 2197, 648, 12, 28

Offset: 1

Seiichi Manyama, Jun 17 2019

			Square array begins:
    1,  1,  1,   1,   1,    1,     1, ...
    3,  3,  3,   3,   3,    3,     3, ...
    4,  4,  4,   4,   4,    4,     4, ...
    7,  9, 13,  21,  37,   69,   133, ...
    6,  6,  6,   6,   6,    6,     6, ...
   12, 24, 66, 216, 762, 2784, 10386, ...
    8,  8,  8,   8,   8,    8,     8, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..3 give A000203, A055225, A308688, A308689.

Cf. A294579, A308509, A308694.

T[n_, k_] := DivisorSum[n, #^(k*n/# - k + 1) &]; Table[T[k, n - k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, May 09 2021 *)

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k*x^j)^(1/j^k)).

A(p,k) = p+1 for prime p.

A356561 Expansion of e.g.f. Product_{k>0} 1/(1 - k^3 * x^k)^(1/k^3).

Original entry on oeis.org

1, 1, 4, 18, 204, 1260, 37440, 299880, 11002320, 204860880, 6618628800, 92924647200, 8181137764800, 124123075876800, 7211104918617600, 288085376346768000, 14964000305173920000, 340302035937191328000, 42619767305209750656000

Offset: 0

Seiichi Manyama, Aug 12 2022

nonn

Cf. A294462, A294469, A356530, A356560.

Cf. A265837, A308689.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1-k^3*x^k)^(1/k^3))))

a308689(n) = sumdiv(n, d, d^(3*n/d-2));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, a308689(j)*v[i-j+1]/(i-j)!)); v;

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} A308689(k) * a(n-k)/(n-k)!.

cp's OEIS Frontend

A308690 Square array A(n,k), n >= 1, k >= 0, where A(n,k) = Sum_{d|n} d^(k*n/d - k + 1), read by antidiagonals.

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