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A292068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + j^k*x^j).

%I A292068 #32 Nov 20 2019 02:26:13
%S A292068 1,1,-1,1,-1,0,1,-1,-1,-1,1,-1,-3,-2,1,1,-1,-7,-6,2,-1,1,-1,-15,-20,6,
%T A292068 -1,1,1,-1,-31,-66,20,5,4,-1,1,-1,-63,-212,66,71,40,-1,2,1,-1,-127,
%U A292068 -666,212,605,442,11,18,-2,1,-1,-255,-2060,666,4439,4660,215,226,-22,2
%N A292068 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + j^k*x^j).
%e A292068 Square array begins:
%e A292068     1,  1,  1,   1,   1, ...
%e A292068    -1, -1, -1,  -1,  -1, ...
%e A292068     0, -1, -3,  -7, -15, ...
%e A292068    -1, -2, -6, -20, -66, ...
%e A292068     1,  2,  6,  20,  66, ...
%p A292068 b:= proc(n, i, k) option remember; (m->
%p A292068       `if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
%p A292068       `if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
%p A292068     end:
%p A292068 A:= proc(n, k) option remember; `if`(n=0, 1,
%p A292068       -add(b(n-i$2, k)*A(i, k), i=0..n-1))
%p A292068     end:
%p A292068 seq(seq(A(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Sep 12 2017
%t A292068 b[n_, i_, k_] := b[n, i, k] = If[# < n, 0, If[n == #, i!^k, b[n, i-1, k] + If[i > n, 0, i^k b[n-i, i-1, k]]]]&[i(i+1)/2];
%t A292068 A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k] A[i, k], {i, 0, n-1}]];
%t A292068 Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Nov 20 2019, after _Alois P. Heinz_ *)
%o A292068 (Python)
%o A292068 from sympy.core.cache import cacheit
%o A292068 from sympy import factorial as f
%o A292068 @cacheit
%o A292068 def b(n, i, k):
%o A292068     m=i*(i + 1)/2
%o A292068     return 0 if m<n else f(i)**k if n==m else b(n, i - 1, k) + (0 if i>n else i**k*b(n - i, i - 1, k))
%o A292068 @cacheit
%o A292068 def A(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*A(i, k) for i in range(n)])
%o A292068 for d in range(13): print([A(n, d - n) for n in range(d + 1)]) # _Indranil Ghosh_, Sep 14 2017, after Maple program
%Y A292068 Columns k=0..2 give A081362, A022693, A292165.
%Y A292068 Rows n=0..2 give A000012, (-1)*A000012, (-1)*A000225.
%Y A292068 Main diagonal gives A292072.
%Y A292068 Cf. A292189, A292193.
%K A292068 sign,tabl
%O A292068 0,13
%A A292068 _Seiichi Manyama_, Sep 12 2017