A346520 Number A(n,k) of partitions of the (n+k)-multiset {0,...,0,1,2,...,k} with n 0's into distinct multisets; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 2, 2, 1, 5, 5, 3, 2, 15, 15, 9, 5, 2, 52, 52, 31, 16, 7, 3, 203, 203, 120, 59, 25, 10, 4, 877, 877, 514, 244, 100, 38, 14, 5, 4140, 4140, 2407, 1112, 442, 161, 56, 19, 6, 21147, 21147, 12205, 5516, 2134, 750, 249, 80, 25, 8, 115975, 115975, 66491, 29505, 11147, 3799, 1213, 372, 111, 33, 10
Offset: 0
Examples
A(2,2) = 9: 00|1|2, 001|2, 1|002, 0|01|2, 0|1|02, 01|02, 00|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, ... 1, 3, 9, 31, 120, 514, 2407, 12205, 66491, ... 2, 5, 16, 59, 244, 1112, 5516, 29505, 168938, ... 2, 7, 25, 100, 442, 2134, 11147, 62505, 373832, ... 3, 10, 38, 161, 750, 3799, 20739, 121141, 752681, ... 4, 14, 56, 249, 1213, 6404, 36332, 220000, 1413937, ... 5, 19, 80, 372, 1887, 10340, 60727, 379831, 2516880, ... 6, 25, 111, 539, 2840, 16108, 97666, 629346, 4288933, ... ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
Crossrefs
Programs
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Maple
g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add( `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n) end: 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, g(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..12); -
Mathematica
g[n_] := g[n] = If[n == 0, 1, Sum[g[n - j]*Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n]; 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, g[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, 12}] // Flatten (* Jean-François Alcover, Jul 31 2021, after Alois P. Heinz *)
Comments