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

Author

Alois P. Heinz, Jul 16 2021

Keywords

• nonn
• tabl

Comments

Also number A(n,k) of factorizations of 2^n * Product_{i=1..k} prime(i+1); A(3,1) = 7: 2*2*2*3, 2*3*4, 4*6, 2*2*6, 3*8, 2*12, 24; A(1,2) = 5: 2*3*5, 5*6, 3*10, 2*15, 30.

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

Columns k=0-10 give: A000041, A000070, A082775, A093802, A346857, A346858, A346859, A346860, A346861, A346862, A346863.

Rows n=0+1, 2-10 give: A000110, A035098, A169587, A169588, A346851, A346852, A346853, A346854, A346855, A346856.

Main diagonal gives A346424.

Antidiagonal sums give A346428.

Cf. A000079, A001055, A008277, A048993, A070826, A126442, A129306, A219727, A255903, A292508, A346520.

Programs

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 *)

Formula

A(n,k) = A001055(A000079(n)*A070826(k+1)).

A(n,k) = Sum_{j=0..k} A048993(k,j)*A292508(n,j+1).

A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000041(n-i).