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
Examples
Square array begins: 1, 1, 1, 1, ... 1, 1, 1, 1, ... 2, 5, 17, 65, ... 3, 32, 746, 19748, ... 5, 298, 66418, 16799044, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..52
Crossrefs
Programs
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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 *) -
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
Formula
G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j^(k*j)).