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A292746 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n with exactly k kinds of 1's which are introduced in ascending order.

%I A292746 #30 Dec 16 2020 08:48:30
%S A292746 1,0,1,1,1,1,1,2,3,1,2,3,8,6,1,2,5,19,26,10,1,4,7,43,97,66,15,1,4,11,
%T A292746 93,334,361,141,21,1,7,15,197,1095,1778,1066,267,28,1,8,22,409,3482,
%U A292746 8207,7108,2668,463,36,1,12,30,840,10855,36310,43747,23116,5909,751,45,1
%N A292746 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n with exactly k kinds of 1's which are introduced in ascending order.
%H A292746 Alois P. Heinz, <a href="/A292746/b292746.txt">Rows n = 0..140, flattened</a>
%F A292746 T(n,k) = A292745(n,k) - A292745(n,k-1) for k>0. T(n,0) = A292745(n,0) = A002865(n).
%F A292746 T(n,k) = Sum_{i=0..k} (-1)^i * A292741(n, k-i) / ((k-i)!*i!).
%e A292746 T(3,0) = 1: 3.
%e A292746 T(3,1) = 2: 21a, 1a1a1a.
%e A292746 T(3,2) = 3: 1a1a1b, 1a1b1a, 1a1b1b. (The two kinds of 1's are labeled 1a and 1b)
%e A292746 T(3,3) = 1: 1a1b1c.
%e A292746 Triangle T(n,k) begins:
%e A292746   1;
%e A292746   0,  1;
%e A292746   1,  1,   1;
%e A292746   1,  2,   3,    1;
%e A292746   2,  3,   8,    6,    1;
%e A292746   2,  5,  19,   26,   10,    1;
%e A292746   4,  7,  43,   97,   66,   15,    1;
%e A292746   4, 11,  93,  334,  361,  141,   21,   1;
%e A292746   7, 15, 197, 1095, 1778, 1066,  267,  28,  1;
%e A292746   8, 22, 409, 3482, 8207, 7108, 2668, 463, 36, 1;
%e A292746   ...
%p A292746 f:= (n, k)-> add(Stirling2(n, j), j=0..k):
%p A292746 b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
%p A292746       f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
%p A292746     end:
%p A292746 T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
%p A292746 seq(seq(T(n, k), k=0..n), n=0..14);
%p A292746 # second Maple program:
%p A292746 b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
%p A292746       k^n, b(n, i-1, k) +b(n-i, min(i, n-i), k))
%p A292746     end:
%p A292746 T:= (n, k)-> add((-1)^i*b(n$2, k-i)/((k-i)!*i!), i=0..k):
%p A292746 seq(seq(T(n, k), k=0..n), n=0..14);
%t A292746 f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
%t A292746 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, f[n, k], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
%t A292746 T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
%t A292746 Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 17 2018, translated from Maple *)
%Y A292746 Columns k=0-10 give: A002865, A000041(n-1) for n>0, A259401(n-2) for n>1, A320816, A320817, A320818, A320819, A320820, A320821, A320822, A320823.
%Y A292746 Main diagonal and first lower diagonal give: A000012, A000217.
%Y A292746 Row sums give A292503.
%Y A292746 T(2n,n) gives A292747.
%Y A292746 Cf. A292741, A292745.
%K A292746 nonn,tabl
%O A292746 0,8
%A A292746 _Alois P. Heinz_, Sep 22 2017
%E A292746 Definition clarified by _N. J. A. Sloane_, Dec 12 2020