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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
 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, 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, 33, 29, 24, 19, 15, 12, 1, 10, 37, 65, 65, 55, 49, 41, 33, 26, 20, 14 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened EXAMPLE A(3,4) =  9: 3, 21a, 21b, 21c, 21d, 1a1b1c, 1a1b1d, 1a1c1d, 1b1c1d. A(4,3) =  8: 4, 31a, 31b, 31c, 22, 21a1b, 21a1c, 21b1c. A(4,4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d. Square array A(n,k) begins:   1,  1,  1,  1,  1,  1,   1,   1,   1, ...   0,  1,  2,  3,  4,  5,   6,   7,   8, ...   1,  1,  2,  4,  7, 11,  16,  22,  29, ...   1,  2,  3,  5,  9, 16,  27,  43,  65, ...   2,  3,  5,  8, 13, 22,  38,  65, 108, ...   2,  4,  7, 12, 20, 33,  55,  93, 158, ...   4,  6, 10, 17, 29, 49,  82, 137, 230, ...   4,  8, 14, 24, 41, 70, 119, 201, 338, ...   7, 11, 19, 33, 57, 98, 168, 287, 488, ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0 or i=1,       binomial(k, n), `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))     end: A:= (n, k)-> b(n\$2, k): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA 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]]; A[n_, k_] := b[n, n, k]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A002865, A027336, A320689, A320690, A320691, A320692, A320693, A320694, A320695, A320696, A320697. Rows n=0-4 give: A000012, A001477, A000124(k-1) for k>0, A011826 for k>0. Main diagonal gives A292507. Cf. A292508, A292741, A292745. Sequence in context: A304382 A304717 A110963 * A292869 A106348 A161092 Adjacent sequences:  A292619 A292620 A292621 * A292623 A292624 A292625 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 20 2017 STATUS approved

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Last modified October 21 20:44 EDT 2019. Contains 328315 sequences. (Running on oeis4.)