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A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

%I A061199 #17 Jan 26 2023 19:59:28
%S A061199 1,0,1,0,1,2,0,2,2,3,0,2,4,4,5,0,3,5,6,6,7,0,4,7,9,10,10,11,0,5,9,12,
%T A061199 13,14,14,15,0,6,13,16,19,20,21,21,22,0,8,16,22,25,27,28,29,29,30,0,
%U A061199 10,22,29,34,37,39,40,41,41,42,0,12,27,38,44,49,51,53,54,55,55,56,0,15,36
%N A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).
%H A061199 Alois P. Heinz, <a href="/A061199/b061199.txt">Columns k = 0..140, flattened</a>
%e A061199 T(2,4) = 4 since the possible partitions of 4 are on the first definition (no term more than twice) 1+1+2, 2+2, 1+3, or 4 and on the second definition (no term a multiple of 3) 1+1+1+1, 1+1+2, 2+2, or 4.
%e A061199 Triangle T(n,k) begins:
%e A061199 1, 0, 0, 0, 0, 0,  0,  0,  0,  0, ...
%e A061199    1, 1, 2, 2, 3,  4,  5,  6,  8, ...
%e A061199       2, 2, 4, 5,  7,  9, 13, 16, ...
%e A061199          3, 4, 6,  9, 12, 16, 22, ...
%e A061199             5, 6, 10, 13, 19, 25, ...
%e A061199                7, 10, 14, 20, 27, ...
%e A061199                   11, 14, 21, 28, ...
%e A061199                       15, 21, 29, ...
%e A061199                           22, 29, ...
%e A061199                               30, ...
%p A061199 b:= proc(n, i, k) option remember;
%p A061199       `if`(n=0, 1, `if`(i<1, 0,
%p A061199       add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
%p A061199     end:
%p A061199 T:= (n, k)-> b(k$2, n):
%p A061199 seq(seq(T(n, k), n=0..k), k=0..12);  # _Alois P. Heinz_, Nov 27 2013
%t A061199 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1, k], {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[k, k, n]; Table[Table[T[n, k], {n, 0, k}], {k, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 28 2015, after _Alois P. Heinz_ *)
%Y A061199 Rows effectively include A000007, A000009, A000726, A001935, A035959.
%Y A061199 Main diagonal is A000041.
%Y A061199 A061198 is the same table but includes cases where n>k.
%Y A061199 T(n,2*n) gives: A232623.
%K A061199 nonn,tabl
%O A061199 0,6
%A A061199 _Henry Bottomley_, Apr 20 2001