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

0,4

Comments

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

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Formula

A(n,k) = A045778(A000079(n)*A070826(k+1)).

A(n,k) = Sum_{j=0..k} Stirling2(k,j)*Sum_{i=0..n} binomial(j+i-1,i)*A000009(n-i).

Example

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, ...
  ...

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 *)

Crossrefs

Columns k=0-10 give: A000009, A036469, A346822, A346823, A346824, A346825, A346826, A346827, A346828, A346829, A346830.

Rows n=0+1,2-10 give: A000110, A087648, A346813, A346814, A346815, A346816, A346817, A346818, A346819, A346820.

Main diagonal gives A346519.

Antidiagonal sums give A346521.

Cf. A000040, A000079, A045778, A048993, A070826, A346426.

Sequence in context: A108087 A123158 A185414 * A362925 A133611 A010094

Adjacent sequences: A346517 A346518 A346519 * A346521 A346522 A346523

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 21 2021

Status

approved