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A292508 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.

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