A275416 Triangle read by rows: T(n,k) is the number of multisets of k odd numbers with a cap of the total sum set to n.

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 4, 3, 1, 1, 3, 8, 5, 3, 1, 1, 4, 10, 10, 5, 3, 1, 1, 4, 16, 15, 11, 5, 3, 1, 1, 5, 20, 27, 17, 11, 5, 3, 1, 1, 5, 29, 38, 32, 18, 11, 5, 3, 1, 1, 6, 35, 60, 49, 34, 18, 11, 5, 3, 1, 1, 6, 47, 84, 83, 54, 35, 18, 11, 5, 3, 1, 1, 7, 56, 122, 123

Offset: 1

Author

R. J. Mathar, Jul 27 2016

Keywords

• nonn
• tabl

Comments

By considering the partitions of n into k parts we set a cap on the odd numbers of each part and count the multisets (ordered k-tuples) of odd numbers where each number is not larger than the cap of its part.

Multiset transformation of A110654 or A065033.

Examples

			T(6,2) = 3+2+3 = 8 counts {1,1} {1,3}, and {3,3} from taking two odd numbers <= 3; it counts {1,1} and {1,3} from taking an odd number <= 2 and an odd number <= 4; and it counts {1,1}, {1,3} and {1,5} from taking an odd number <= 1 and an odd number <= 5.
T(6,3) = 1+2+2 = 5 counts {1,1,1} from taking three odd numbers <= 2; it counts {1,1,1} and {1,1,3} from taking an odd number <= 1 and an odd number <= 2 and an odd number <= 3; and it counts {1,1,1} and {1,1,3} from taking two odd numbers <= 1 and an odd number <= 4.
  1
  1   1
  2   1   1
  2   3   1   1
  3   4   3   1   1
  3   8   5   3   1   1
  4  10  10   5   3   1   1
  4  16  15  11   5   3   1   1
  5  20  27  17  11   5   3   1   1
  5  29  38  32  18  11   5   3   1   1
  6  35  60  49  34  18  11   5   3   1   1
  6  47  84  83  54  35  18  11   5   3   1   1
  7  56 122 123  94  56  35  18  11   5   3   1   1
  7  72 164 192 146  99  57  35  18  11   5   3   1   1
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened
• Index entries for triangles generated by the Multiset Transformation

Crossrefs

Cf. A110654 (column 1), A003293 (row sums?), A089353 (equivalent Multiset transformation of A000027), A005232 (2nd column?), A097513 (3rd column?).

T(2n,n) gives A269628.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(ceil(i/2)+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..16);  # Alois P. Heinz, Apr 13 2017

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[Ceiling[i/2] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

Formula

T(n,1) = A110654(n).

T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1 < k <= n.

G.f.: Product_{j>=1} (1-y*x^j)^(-ceiling(j/2)). - Alois P. Heinz, Apr 13 2017