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Search: id:A144299
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| A144299 |
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Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), ..., T(n,0) for n >= 0. |
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7
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| 1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 3, 0, 0, 1, 10, 15, 0, 0, 0, 1, 15, 45, 15, 0, 0, 0, 1, 21, 105, 105, 0, 0, 0, 0, 1, 28, 210, 420, 105, 0, 0, 0, 0, 1, 36, 378, 1260, 945, 0, 0, 0, 0, 0, 1, 45, 630, 3150, 4725, 945, 0, 0, 0, 0, 0, 1, 55, 990, 6930, 17325, 10395, 0, 0, 0, 0, 0, 0, 1, 66, 1485, 13860, 51975, 62370
(list; table; graph; listen)
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OFFSET
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1,8
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COMMENT
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T(n,k) = number of partitions of an n-set into k nonempty subsets, each of size at most 2.
The Grosswald and Choi-Smith references give many further properties and formulae.
Considered as an infinite lower triangular matrix T, Lim_{n->inf.} T^n = A118930: (1, 1, 2, 4, 13, 41, 166, 652, ...) as a vector. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 08 2008]
A001498 has a b-file.
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REFERENCES
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J. Y. Choi and J. D. H. Smith, On the unimodilty and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
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LINKS
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David Applegate and N. J. A. Sloane, The Gift Exchange Problem (arXiv:0907.0513, 2009)
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FORMULA
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T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).
E.g.f.: Sum_{k >= 0} Sum_{n = 0..2k} T(n,k) y^k x^n/n! = exp(y(x+x^2/2)). (The coefficient of y^k is the e.g.f. for the k-th row of the rotated triangle shown below.)
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EXAMPLE
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Triangle begins:
n:
0: 1
1: 1 0
2: 1 1 0
3: 1 3 0 0
4: 1 6 3 0 0
5: 1 10 15 0 0 0
6: 1 15 45 15 0 0 0
7: 1 21 105 105 0 0 0 0
8: 1 28 210 420 105 0 0 0 0
9: 1 36 378 1260 945 0 0 0 0 0
...
The row sums give A000085.
For some purposes it is convenient to rotate the triangle by 45 degrees:
1.0.0.0.0..0..0...0...0....0....0.....0....
..1.1.0.0..0..0...0...0....0....0.....0....
....1.3.3..0..0...0...0....0....0.....0....
......1.6.15.15...0...0....0....0.....0....
........1.10.45.105.105....0....0.....0....
...........1.15.105.420..945..945.....0....
..............1..21.210.1260.4725.10395....
..................1..28..378.3150.17325....
......................1...36..630..6930....
...........................1...45...990....
...
The latter triangle is important enough that it has its own entry, A144331. Here the column sums give A000085 and the rows sums give A001515.
If the entries in the rotated triangle are denoted by b1(n,k), n >= 0, k <= 2n, the we have the recurrence b1(n, k) = b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2).
Then b1(n,k) is the number of partitions of [1, 2, ...,k] into exactly n blocks, each of size 1 or 2.
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MAPLE
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Maple code producing the rotated version:
b1 := proc(n, k)
option remember;
if n = k then 1;
elif k < n then 0;
elif n < 1 then 0;
else b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2);
end if;
end proc;
for n from 0 to 12 do lprint([seq(b1(n, k), k=0..2*n)]); od:
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MATHEMATICA
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T[n_, 0] = 0; T[1, 1] = 1; T[2, 1] = 1; T[n_, k_] := T[n - 1, k - 1] + (n - 1)T[n - 2, k - 1]; Table[T[n, k], {n, 12}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)
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CROSSREFS
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Other versions of this same triangle are given in A111924 (which omits the first row), A001498 (which left-adjusts the rows in the bottom view), A001497 and A100861. Row sums give A000085.
Cf. A144385, A144643.
Cf. A118930 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 08 2008]
Sequence in context: A126723 A090030 A080159 this_sequence A060514 A096936 A115979
Adjacent sequences: A144296 A144297 A144298 this_sequence A144300 A144301 A144302
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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David Applegate and N. J. A. Sloane (njas(AT)research.att.com), Dec 06 2008
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