A275416 - OEIS

%I #30 Dec 22 2018 03:32:34

%S 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,

%T 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,

%U 18,11,5,3,1,1,6,47,84,83,54,35,18,11,5,3,1,1,7,56,122,123

%N 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.

%C 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.

%C Multiset transformation of A110654 or A065033.

%H Alois P. Heinz, <a href="/A275416/b275416.txt">Rows n = 1..200, flattened</a>

%H <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>

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

%F 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.

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

%e 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.

%e 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.

%e   1

%e   1   1

%e   2   1   1

%e   2   3   1   1

%e   3   4   3   1   1

%e   3   8   5   3   1   1

%e   4  10  10   5   3   1   1

%e   4  16  15  11   5   3   1   1

%e   5  20  27  17  11   5   3   1   1

%e   5  29  38  32  18  11   5   3   1   1

%e   6  35  60  49  34  18  11   5   3   1   1

%e   6  47  84  83  54  35  18  11   5   3   1   1

%e   7  56 122 123  94  56  35  18  11   5   3   1   1

%e   7  72 164 192 146  99  57  35  18  11   5   3   1   1

%p b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,

%p       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*

%p        binomial(ceil(i/2)+j-1, j), j=0..min(n/i, p)))))

%p     end:

%p T:= (n, k)-> b(n$2, k):

%p seq(seq(T(n, k), k=1..n), n=1..16);  # _Alois P. Heinz_, Apr 13 2017

%t 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 T[n_, k_] := b[n, n, k];

%t Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 19 2018, after _Alois P. Heinz_ *)

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

%Y T(2n,n) gives A269628.

%K nonn,tabl

%O 1,4

%A _R. J. Mathar_, Jul 27 2016