A283674 - OEIS

%I #30 Mar 17 2017 10:09:09

%S 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,

%T 3531,11,1,1,1025,531698,16799044,9843707,51609,15,1,1,4097,14349932,

%U 4295531890,30535636881,2187941520,894834,22,1,1,16385,387424586,1099526502508,95371863221411,101591759812967,680615139257,17980052,30

%N 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.

%H Alois P. Heinz, <a href="/A283674/b283674.txt">Antidiagonals n = 0..52</a>

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

%e Square array begins:

%e    1,   1,     1,        1, ...

%e    1,   1,     1,        1, ...

%e    2,   5,    17,       65, ...

%e    3,  32,   746,    19748, ...

%e    5, 298, 66418, 16799044, ...

%p with(numtheory):

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(add(

%p       d*d^(k*d), d=divisors(j))*A(n-j, k), j=1..n)/n)

%p     end:

%p seq(seq(A(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Mar 15 2017

%t 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 *)

%o (PARI) A(n, k) = if(n==0, 1, sum(j=1, n, sumdiv(j, d, d*d^(k*d)) * A(n - j, k))/n);

%o {for(d=0, 10, for(n=0, d, print1(A(n, d - n),", ");); print(););} \\ _Indranil Ghosh_, Mar 17 2017

%Y Columns k=0-4 give A000041, A023880, A283579, A283580, A283510.

%Y Rows give: 0-1: A000012, 2: A052539, 3: A283716.

%Y Main diagonal gives A283719.

%Y Cf. A283675.

%K nonn,tabl

%O 0,6

%A _Seiichi Manyama_, Mar 14 2017