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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.

%I A275416 #30 Dec 22 2018 03:32:34
%S A275416 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 A275416 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 A275416 18,11,5,3,1,1,6,47,84,83,54,35,18,11,5,3,1,1,7,56,122,123
%N 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.
%C A275416 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 A275416 Multiset transformation of A110654 or A065033.
%H A275416 Alois P. Heinz, <a href="/A275416/b275416.txt">Rows n = 1..200, flattened</a>
%H A275416 <a href="/index/Mu#multiplicative_completely">Index entries for triangles generated by the Multiset Transformation</a>
%F A275416 T(n,1) = A110654(n).
%F A275416 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 A275416 G.f.: Product_{j>=1} (1-y*x^j)^(-ceiling(j/2)). - _Alois P. Heinz_, Apr 13 2017
%e A275416 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 A275416 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 A275416   1
%e A275416   1   1
%e A275416   2   1   1
%e A275416   2   3   1   1
%e A275416   3   4   3   1   1
%e A275416   3   8   5   3   1   1
%e A275416   4  10  10   5   3   1   1
%e A275416   4  16  15  11   5   3   1   1
%e A275416   5  20  27  17  11   5   3   1   1
%e A275416   5  29  38  32  18  11   5   3   1   1
%e A275416   6  35  60  49  34  18  11   5   3   1   1
%e A275416   6  47  84  83  54  35  18  11   5   3   1   1
%e A275416   7  56 122 123  94  56  35  18  11   5   3   1   1
%e A275416   7  72 164 192 146  99  57  35  18  11   5   3   1   1
%p A275416 b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
%p A275416       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
%p A275416        binomial(ceil(i/2)+j-1, j), j=0..min(n/i, p)))))
%p A275416     end:
%p A275416 T:= (n, k)-> b(n$2, k):
%p A275416 seq(seq(T(n, k), k=1..n), n=1..16);  # _Alois P. Heinz_, Apr 13 2017
%t A275416 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 A275416 T[n_, k_] := b[n, n, k];
%t A275416 Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 19 2018, after _Alois P. Heinz_ *)
%Y A275416 Cf. A110654 (column 1), A003293 (row sums?), A089353 (equivalent Multiset transformation of A000027), A005232 (2nd column?), A097513 (3rd column?).
%Y A275416 T(2n,n) gives A269628.
%K A275416 nonn,tabl
%O A275416 1,4
%A A275416 _R. J. Mathar_, Jul 27 2016