A346426 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 2, 2, 2, 5, 5, 4, 3, 15, 15, 11, 7, 5, 52, 52, 36, 21, 12, 7, 203, 203, 135, 74, 38, 19, 11, 877, 877, 566, 296, 141, 64, 30, 15, 4140, 4140, 2610, 1315, 592, 250, 105, 45, 22, 21147, 21147, 13082, 6393, 2752, 1098, 426, 165, 67, 30, 115975, 115975, 70631, 33645, 13960, 5317, 1940, 696, 254, 97, 42
Offset: 0
Examples
A(2,2) = 11: 00|1|2, 001|2, 1|002, 0|0|1|2, 0|01|2, 0|1|02, 01|02, 00|12, 0|0|12, 0|012, 0012. Square array A(n,k) begins: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ... 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ... 2, 4, 11, 36, 135, 566, 2610, 13082, 70631, ... 3, 7, 21, 74, 296, 1315, 6393, 33645, 190085, ... 5, 12, 38, 141, 592, 2752, 13960, 76464, 448603, ... 7, 19, 64, 250, 1098, 5317, 28009, 158926, 963913, ... 11, 30, 105, 426, 1940, 9722, 52902, 309546, 1933171, ... 15, 45, 165, 696, 3281, 16972, 95129, 572402, 3670878, ... 22, 67, 254, 1106, 5372, 28582, 164528, 1015356, 6670707, ... ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Crossrefs
Programs
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Maple
s:= proc(n) option remember; expand(`if`(n=0, 1, x*add(s(n-j)*binomial(n-1, j-1), j=1..n))) end: S:= proc(n, k) option remember; coeff(s(n), x, k) end: b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=0, combinat[numbpart](n), add(b(n-j, i-1), j=0..n))) end: A:= (n, k)-> add(S(k, j)*b(n, j), j=0..k): seq(seq(A(n, d-n), n=0..d), d=0..10); -
Mathematica
s[n_] := s[n] = Expand[If[n == 0, 1, x Sum[s[n - j] Binomial[n - 1, j - 1], {j, 1, n}]]]; S[n_, k_] := S[n, k] = Coefficient[s[n], x, k]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, PartitionsP[n], Sum[b[n - j, i - 1], {j, 0, n}]]]; A[n_, k_] := Sum[S[k, j] b[n, j], {j, 0, k}]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Aug 18 2021, after Alois P. Heinz *)
Comments