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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
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
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Mar 14 2017
STATUS
approved