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A292622 Number A(n,k) of partitions of n with up to k distinct kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A292622 #30 Apr 25 2021 03:02:14
%S A292622 1,1,0,1,1,1,1,2,1,1,1,3,2,2,2,1,4,4,3,3,2,1,5,7,5,5,4,4,1,6,11,9,8,7,
%T A292622 6,4,1,7,16,16,13,12,10,8,7,1,8,22,27,22,20,17,14,11,8,1,9,29,43,38,
%U A292622 33,29,24,19,15,12,1,10,37,65,65,55,49,41,33,26,20,14
%N A292622 Number A(n,k) of partitions of n with up to k distinct kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A292622 For fixed k>=0, A(n,k) ~ Pi * 2^(k - 5/2) * exp(Pi*sqrt(2*n/3)) / (3 * n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2018
%H A292622 Alois P. Heinz, <a href="/A292622/b292622.txt">Antidiagonals n = 0..140, flattened</a>
%F A292622 G.f. of column k: (1 + x)^k * Product_{j>=2} 1 / (1 - x^j). - _Ilya Gutkovskiy_, Apr 24 2021
%e A292622 A(3,4) =  9: 3, 21a, 21b, 21c, 21d, 1a1b1c, 1a1b1d, 1a1c1d, 1b1c1d.
%e A292622 A(4,3) =  8: 4, 31a, 31b, 31c, 22, 21a1b, 21a1c, 21b1c.
%e A292622 A(4,4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d.
%e A292622 Square array A(n,k) begins:
%e A292622   1,  1,  1,  1,  1,  1,   1,   1,   1, ...
%e A292622   0,  1,  2,  3,  4,  5,   6,   7,   8, ...
%e A292622   1,  1,  2,  4,  7, 11,  16,  22,  29, ...
%e A292622   1,  2,  3,  5,  9, 16,  27,  43,  65, ...
%e A292622   2,  3,  5,  8, 13, 22,  38,  65, 108, ...
%e A292622   2,  4,  7, 12, 20, 33,  55,  93, 158, ...
%e A292622   4,  6, 10, 17, 29, 49,  82, 137, 230, ...
%e A292622   4,  8, 14, 24, 41, 70, 119, 201, 338, ...
%e A292622   7, 11, 19, 33, 57, 98, 168, 287, 488, ...
%p A292622 b:= proc(n, i, k) option remember; `if`(n=0 or i=1,
%p A292622       binomial(k, n), `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))
%p A292622     end:
%p A292622 A:= (n, k)-> b(n$2, k):
%p A292622 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A292622 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, Binomial[k, n], If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
%t A292622 A[n_, k_] := b[n, n, k];
%t A292622 Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, May 19 2018, after _Alois P. Heinz_ *)
%Y A292622 Columns k=0-10 give: A002865, A027336, A320689, A320690, A320691, A320692, A320693, A320694, A320695, A320696, A320697.
%Y A292622 Rows n=0-4 give: A000012, A001477, A000124(k-1) for k>0, A011826 for k>0.
%Y A292622 Main diagonal gives A292507.
%Y A292622 Cf. A292508, A292741, A292745.
%K A292622 nonn,tabl
%O A292622 0,8
%A A292622 _Alois P. Heinz_, Sep 20 2017