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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. 16

%I

%S 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 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 33,29,24,19,15,12,1,10,37,65,65,55,49,41,33,26,20,14

%N 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 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 Alois P. Heinz, <a href="/A292622/b292622.txt">Antidiagonals n = 0..140, flattened</a>

%e A(3,4) = 9: 3, 21a, 21b, 21c, 21d, 1a1b1c, 1a1b1d, 1a1c1d, 1b1c1d.

%e A(4,3) = 8: 4, 31a, 31b, 31c, 22, 21a1b, 21a1c, 21b1c.

%e A(4,4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d.

%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, 1, 2, 4, 7, 11, 16, 22, 29, ...

%e 1, 2, 3, 5, 9, 16, 27, 43, 65, ...

%e 2, 3, 5, 8, 13, 22, 38, 65, 108, ...

%e 2, 4, 7, 12, 20, 33, 55, 93, 158, ...

%e 4, 6, 10, 17, 29, 49, 82, 137, 230, ...

%e 4, 8, 14, 24, 41, 70, 119, 201, 338, ...

%e 7, 11, 19, 33, 57, 98, 168, 287, 488, ...

%p b:= proc(n, i, k) option remember; `if`(n=0 or i=1,

%p binomial(k, n), `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))

%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 == 1, Binomial[k, n], If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];

%t A[n_, k_] := b[n, n, k];

%t Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-Fran├žois Alcover_, May 19 2018, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A002865, A027336, A320689, A320690, A320691, A320692, A320693, A320694, A320695, A320696, A320697.

%Y Rows n=0-4 give: A000012, A001477, A000124(k-1) for k>0, A011826 for k>0.

%Y Main diagonal gives A292507.

%Y Cf. A292508, A292741, A292745.

%K nonn,tabl

%O 0,8

%A _Alois P. Heinz_, Sep 20 2017

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Last modified November 21 01:33 EST 2019. Contains 329349 sequences. (Running on oeis4.)