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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.
13
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, 93, 334, 361, 141, 21, 1, 7, 15, 197, 1095, 1778, 1066, 267, 28, 1, 8, 22, 409, 3482, 8207, 7108, 2668, 463, 36, 1, 12, 30, 840, 10855, 36310, 43747, 23116, 5909, 751, 45, 1
OFFSET
0,8
LINKS
FORMULA
T(n,k) = A292745(n,k) - A292745(n,k-1) for k>0. T(n,0) = A292745(n,0) = A002865(n).
T(n,k) = Sum_{i=0..k} (-1)^i * A292741(n, k-i) / ((k-i)!*i!).
EXAMPLE
T(3,0) = 1: 3.
T(3,1) = 2: 21a, 1a1a1a.
T(3,2) = 3: 1a1a1b, 1a1b1a, 1a1b1b. (The two kinds of 1's are labeled 1a and 1b)
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
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, 93, 334, 361, 141, 21, 1;
7, 15, 197, 1095, 1778, 1066, 267, 28, 1;
8, 22, 409, 3482, 8207, 7108, 2668, 463, 36, 1;
...
MAPLE
f:= (n, k)-> add(Stirling2(n, j), j=0..k):
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
end:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=0..14);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
k^n, b(n, i-1, k) +b(n-i, min(i, n-i), k))
end:
T:= (n, k)-> add((-1)^i*b(n$2, k-i)/((k-i)!*i!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..14);
MATHEMATICA
f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
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[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 17 2018, translated from Maple *)
CROSSREFS
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.
Main diagonal and first lower diagonal give: A000012, A000217.
Row sums give A292503.
T(2n,n) gives A292747.
Sequence in context: A309940 A081514 A109206 * A176506 A200596 A088422
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 22 2017
EXTENSIONS
Definition clarified by N. J. A. Sloane, Dec 12 2020
STATUS
approved