A346822 -id:A346822 - cp's OEIS frontend

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

Alois P. Heinz, Jul 21 2021

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.

			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, ...
  ...

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

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.

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);

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 *)

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).

A082775 Convolution of natural numbers >= 2 and the partition numbers (A000041).

Original entry on oeis.org

2, 5, 11, 21, 38, 64, 105, 165, 254, 381, 562, 813, 1162, 1636, 2279, 3139, 4285, 5794, 7776, 10353, 13694, 17992, 23502, 30520, 39433, 50687, 64855, 82607, 104785, 132375, 166608, 208921, 261090, 325196, 403779, 499818, 616928, 759335, 932135

Offset: 2

Alford Arnold, May 22 2003

Contribution from George Beck, Jan 08 2011: (Start)
The number of multiset partitions of the n-multiset M={0,0,...,0,1,2} (with n-2 zeros) is sum_{k=0..(n-2)}( (n-k) * p(k) ) where p(k) is the number of partitions of k.
Proof:
For each k = 0, 1, ..., n-2, partition k zeros and add the remaining n-k-2 zeros to the block {1, 2}, to give p(k) partitions.
For each k, partition k zeros and add the remaining n-k-2 zeros to the two blocks {1} and {2} in all possible 1 + n-k-2 ways, which gives (1 + n-k-2) * p(k) partitions.
Together, the number of partitions of M is sum_{k=0..n-2}( (n-k) * p(k) ). (End)
A082775 is the special case of A126442 with n-k = 2.

			a(7) = 64 because (7,5,3,2,1,1) dot (2,3,4,5,6,7) = 14+15+12+10+6+7= 64.

Cf. A023548, A126442, A346822.

Column k=2 of A346426.

f[n_] := Sum[(n - k) PartitionsP[k], {k, 0, n - 2}]; Array[f, 39, 2]

a(n) = a(n-1) + A000041(n) + A000070(n) for n>1. - Alford Arnold, Dec 10 2007
a(n) = n*A000070(n-2) - A182738(n-2) for n>2. - Vaclav Kotesovec, Jun 23 2015
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - Vaclav Kotesovec, Jun 23 2015

More terms from Ray Chandler, Oct 11 2003

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula