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%I #17 Feb 11 2023 11:33:35
%S 1,0,1,0,1,1,0,1,1,1,0,2,2,1,1,0,2,2,2,1,1,0,3,4,3,2,1,1,0,4,5,4,3,2,
%T 1,1,0,5,7,6,5,3,2,1,1,0,6,9,9,6,5,3,2,1,1,0,8,13,12,10,7,5,3,2,1,1,0,
%U 10,16,16,13,10,7,5,3,2,1,1,0,12,22,22,19,14,11,7,5,3,2,1,1,0,15,27,29,25,20,14,11,7,5,3,2,1,1
%N Square table by antidiagonals where T(n,k) is number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).
%F G.f. for row n of table: Product_{j>=1} Sum_{k=0..n} x^(j*k) = Product_{j>=1} (1-x^((n+1)*j)) / (1-x^j). - _Sean A. Irvine_, Jan 26 2023
%e Square table T(n,k) begins:
%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ...
%e 1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, ...
%e 1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, ...
%e 1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ...
%e 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ...
%e 1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, ...
%e 1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, ...
%e 1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, ...
%e 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, ...
%e 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
%p end:
%p A:= (n, k)-> b(k$2, n):
%p seq(seq(A(n, d-n), n=0..d), d=0..13); # _Alois P. Heinz_, Jan 26 2023
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, k], {j, 0, Min[n/i, k]}]]];
%t A[n_, k_] := b[k, k, n];
%t Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* _Jean-François Alcover_, Feb 11 2023, after _Alois P. Heinz_ *)
%Y Rows include A000007, A000009, A000726, A035959.
%Y Main diagonal is A000041.
%Y A061199 is the same table but excluding cases where n>k.
%Y Cf. A286653.
%K nonn,tabl
%O 0,12
%A _Henry Bottomley_, Apr 20 2001
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