%I #40 Jul 16 2021 20:56:38
%S 1,1,0,1,1,1,1,2,2,1,1,3,4,3,2,1,4,7,7,5,2,1,5,11,14,12,7,4,1,6,16,25,
%T 26,19,11,4,1,7,22,41,51,45,30,15,7,1,8,29,63,92,96,75,45,22,8,1,9,37,
%U 92,155,188,171,120,67,30,12,1,10,46,129,247,343,359,291,187,97,42,14
%N Number A(n,k) of partitions of n with k kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C Partial sum operator applied to column k gives column k+1.
%C A(n,k) is also defined for k < 0. All given formulas and programs can be applied also if k is negative.
%H Alois P. Heinz, <a href="/A292508/b292508.txt">Antidiagonals n = 0..140, flattened</a>
%F G.f. of column k: 1/(1-x)^k * 1/Product_{j>1} (1-x^j).
%F Column k is Euler transform of k,1,1,1,... .
%F For fixed k>=0, A(n,k) ~ 2^((k-5)/2) * 3^((k-2)/2) * n^((k-3)/2) * exp(Pi*sqrt(2*n/3)) / Pi^(k-1). - _Vaclav Kotesovec_, Oct 24 2018
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
%e 1, 2, 4, 7, 11, 16, 22, 29, 37, ...
%e 1, 3, 7, 14, 25, 41, 63, 92, 129, ...
%e 2, 5, 12, 26, 51, 92, 155, 247, 376, ...
%e 2, 7, 19, 45, 96, 188, 343, 590, 966, ...
%e 4, 11, 30, 75, 171, 359, 702, 1292, 2258, ...
%e 4, 15, 45, 120, 291, 650, 1352, 2644, 4902, ...
%e 7, 22, 67, 187, 478, 1128, 2480, 5124, 10026, ...
%p A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p (numtheory[sigma](j)+k-1)*A(n-j, k), j=1..n)/n)
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%p # second Maple program:
%p A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k<1,
%p A(n, k+1)-A(n-1, k+1), `if`(k=1, combinat[numbpart](n),
%p A(n-1, k)+A(n, k-1))))
%p end:
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%p # third Maple program:
%p b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
%p binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
%p end:
%p A:= (n, k)-> b(n$2, k):
%p seq(seq(A(n, d-n), n=0..d), d=0..14);
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Binomial[k + n - 1, n], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
%t A[n_, k_] := b[n, n, k];
%t Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 17 2018, translated from 3rd Maple program *)
%Y Columns k=0-10 give: A002865, A000041, A000070, A014153, A014160, A014161, A120477, A320753, A320754, A320755, A320756.
%Y Rows n=0-4 give: A000012, A001477, A000124, A004006(k+1), A027927(k+3).
%Y Main diagonal gives A292463.
%Y A(n,n+1) gives A292613.
%Y Cf. A292622, A292741, A292745.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Sep 17 2017