A283803 -id:A283803 - cp's OEIS frontend

A283675 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-x^j)^(j^(k*j)) in powers of x.

1, 1, -1, 1, -1, -1, 1, -1, -4, 0, 1, -1, -16, -23, 0, 1, -1, -64, -713, -223, 1, 1, -1, -256, -19619, -64687, -2767, 0, 1, -1, -1024, -531185, -16755517, -9688545, -42268, 1, 1, -1, -4096, -14347883, -4294403215, -30499543213, -2165715003, -759008, 0, 1, -1, -16384

Offset: 0

Seiichi Manyama, Mar 14 2017

			Square array begins:
   1,    1,      1,         1, ...
  -1,   -1,     -1,        -1, ...
  -1,   -4,    -16,       -64, ...
   0,  -23,   -713,    -19619, ...
   0, -223, -64687, -16755517, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..4 give A010815, A283499, A283534, A283536, A283803.
Rows n=0..1 give A000012, (-1)*A000012.
Main diagonal gives A283720.
Cf. A283674.

G.f. of column k: Product_{j>=1} (1-x^j)^(j^(k*j)).

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k*d+1)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 04 2017

A283510 Expansion of exp( Sum_{n>=1} A283369(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, 1, 257, 531698, 4295531890, 95371863221411, 4738477950914329100, 459991301719292572342573, 79228623778497392212453912974, 22528478894247280128054776211273960, 10000022549030658394108744658459680045250

Offset: 0

Seiichi Manyama, Mar 17 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..120

Cf. Product_{k>=1} 1/(1 - x^k)^(k^(m*k)): A000041 (m=0), A023880 (m=1), A283579 (m=2), A283580 (m=3), this sequence (m=4).

Cf. A283803 (Product_{k>=1} (1 - x^k)^(k^(4*k))).

CoefficientList[Series[Product[1/(1 - x^k)^(k^(4k)), {k, 1, 10}], {x, 0, 10}], x] (* Indranil Ghosh, Mar 17 2017 *)

A(n) = sumdiv(n, d, d^(4*d + 1));
a(n) = if(n<1, 1, (1/n) * sum(k=1, n, A(k) * a(n - k)));
for(n=0, 10, print1(a(n),", ")) \\ Indranil Ghosh, Mar 17 2017

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^(4*k)).
a(n) = (1/n)*Sum_{k=1..n} A283369(k)*a(n-k) for n > 0.
a(n) ~ n^(4*n) * (1 + exp(-4)/n^4). - Vaclav Kotesovec, Mar 17 2017

cp's OEIS Frontend

A283675 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-x^j)^(j^(k*j)) in powers of x.

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