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

0,6

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

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.

Sequence in context: A117396 A125860 A294585 * A294758 A125800 A264698

Adjacent sequences: A283671 A283672 A283673 * A283675 A283676 A283677

Keyword

nonn,tabl

Author

Seiichi Manyama, Mar 14 2017

Status

approved