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
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
Crossrefs
Programs
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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
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