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A256161 Triangle of allowable Stirling numbers of the second kind a(n,k). 2
1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 6, 1, 1, 5, 26, 23, 9, 1, 1, 6, 57, 72, 50, 12, 1, 1, 7, 120, 201, 222, 86, 16, 1, 1, 8, 247, 522, 867, 480, 150, 20, 1, 1, 9, 502, 1291, 3123, 2307, 1080, 230, 25, 1, 1, 10, 1013, 3084, 10660, 10044, 6627, 2000, 355, 30, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Row sums = A007476 starting (1, 2, 4, 9, 23, 65, 199, 654, 2296, 8569, ...).
a(n,k) counts restricted growth words of length n in the letters {1, ..., k} where every even entry appears exactly once.
LINKS
Yue Cai and Margaret Readdy, Negative q-Stirling numbers, arXiv:1506.03249 [math.CO], 2015.
FORMULA
a(n,k) = a(n-1,k-1) + ceiling(k/2)*a(n-1,k) for n >= 1 and 1 <= k <= n with boundary conditions a(n,0) = KroneckerDelta[n,0].
a(n,2) = n-1.
a(n,n-1) = floor(n/2)*ceiling(n/2).
EXAMPLE
a(4,1) = 1 via 1111;
a(4,2) = 3 via 1211, 1121, 1112;
a(4,3) = 4 via 1213, 1231, 1233, 1123;
a(4,4) = 1 via 1234.
Triangle starts:
1;
1, 1;
1, 2, 1;
1, 3, 4, 1;
1, 4, 11, 6, 1;
...
MATHEMATICA
a[_, 1] = a[n_, n_] = 1;
a[n_, k_] := a[n, k] = a[n-1, k-1] + Ceiling[k/2] a[n-1, k];
Table[a[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 15 2018 *)
CROSSREFS
Cf. A007476 (row sums), A246118 (essentially the same triangle).
Sequence in context: A137153 A340814 A063841 * A137596 A111669 A336573
KEYWORD
nonn,tabl
AUTHOR
Margaret A. Readdy, Mar 16 2015
STATUS
approved

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Last modified February 29 11:59 EST 2024. Contains 370425 sequences. (Running on oeis4.)