A283510 -id:A283510 - cp's OEIS frontend

A283674 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-x^j)^(j^(k*j)) in powers of x.

1, 1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 32, 5, 1, 1, 65, 746, 298, 7, 1, 1, 257, 19748, 66418, 3531, 11, 1, 1, 1025, 531698, 16799044, 9843707, 51609, 15, 1, 1, 4097, 14349932, 4295531890, 30535636881, 2187941520, 894834, 22, 1, 1, 16385, 387424586, 1099526502508, 95371863221411, 101591759812967, 680615139257, 17980052, 30

Offset: 0

Seiichi Manyama, Mar 14 2017

			Square array begins:
   1,   1,     1,        1, ...
   1,   1,     1,        1, ...
   2,   5,    17,       65, ...
   3,  32,   746,    19748, ...
   5, 298, 66418, 16799044, ...

Alois P. Heinz, Antidiagonals n = 0..52

Columns k=0-4 give A000041, A023880, A283579, A283580, A283510.
Rows give: 0-1: A000012, 2: A052539, 3: A283716.
Main diagonal gives A283719.
Cf. A283675.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      d*d^(k*d), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Mar 15 2017

A[n_, k_] := If[n==0, 1, Sum[Sum[d*d^(k*d), {d, Divisors[j]}] *A[n - j, k], {j, n}] / n]; Flatten[Table[A[d - n,  n],{d, 0, 10},{n, d, 0, -1}]] (* Indranil Ghosh, Mar 17 2017 *)

A(n, k) = if(n==0, 1, sum(j=1, n, sumdiv(j, d, d*d^(k*d)) * A(n - j, k))/n);
{for(d=0, 10, for(n=0, d, print1(A(n, d - n),", ");); print(););} \\ Indranil Ghosh, Mar 17 2017

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

A283803 Expansion of exp( Sum_{n>=1} -A283369(n)/n*x^n ) in powers of x.

Original entry on oeis.org

1, -1, -256, -531185, -4294403215, -95363000657073, -4738284730302658391, -459981771468075494207385, -79227701254823507875355278590, -22528320196093613328344381426130010, -9999977451048811940735941180766259658078

Offset: 0

Seiichi Manyama, Mar 17 2017

sign

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

Cf. Product_{k>=1} (1 - x^k)^(k^(m*k)): A010815 (m=0), A283499 (m=1), A283534 (m=2), A283536 (m=3), this sequence (m=4).

Cf. A283510 (Product_{k>=1} 1/(1 - x^k)^(k^(4*k))).

CoefficientList[Series[Product[(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 - x^k)^(k^(4*k)).

a(n) = -(1/n)*Sum_{k=1..n} A283369(k)*a(n-k) for n > 0.

cp's OEIS Frontend

A283674 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1-x^j)^(j^(k*j)) in powers of x.

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