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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified September 16 12:41 EDT 2019. Contains 327113 sequences. (Running on oeis4.)