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A292745
Number A(n,k) of partitions of n with k sorts of part 1 which are introduced in ascending order; square array A(n,k), n>=0, k>=0, read by antidiagonals.
15
1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 3, 6, 5, 2, 1, 1, 3, 7, 13, 7, 4, 1, 1, 3, 7, 19, 26, 11, 4, 1, 1, 3, 7, 20, 52, 54, 15, 7, 1, 1, 3, 7, 20, 62, 151, 108, 22, 8, 1, 1, 3, 7, 20, 63, 217, 442, 219, 30, 12, 1, 1, 3, 7, 20, 63, 232, 803, 1314, 439, 42, 14
OFFSET
0,9
LINKS
FORMULA
A(n,k) = Sum_{j=0..k} A292746(n,j).
A(n,k) = A(n,n) for all k >= n.
EXAMPLE
A(3,2) = 6: 3, 21a, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 3, 3, 3, 3, 3, 3, ...
1, 3, 6, 7, 7, 7, 7, 7, 7, ...
2, 5, 13, 19, 20, 20, 20, 20, 20, ...
2, 7, 26, 52, 62, 63, 63, 63, 63, ...
4, 11, 54, 151, 217, 232, 233, 233, 233, ...
4, 15, 108, 442, 803, 944, 965, 966, 966, ...
7, 22, 219, 1314, 3092, 4158, 4425, 4453, 4454, ...
MAPLE
f:= (n, k)-> add(Stirling2(n, j), j=0..k):
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, f[n, k], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 17 2018, translated from Maple *)
CROSSREFS
Main diagonal gives A292503.
Sequence in context: A263447 A180303 A118923 * A047010 A047100 A124772
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 22 2017
STATUS
approved