Search: id:A144299
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%I A144299
%S A144299 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,
%T A144299 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,
%U A144299 0,0,0,1,55,990,6930,17325,10395,0,0,0,0,0,0,1,66,1485,13860,51975,62370
%N A144299 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.
%C A144299 T(n,k) = number of partitions of an n-set into k nonempty subsets, each 
               of size at most 2.
%C A144299 The Grosswald and Choi-Smith references give many further properties 
               and formulae.
%C A144299 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]
%C A144299 A001498 has a b-file.
%D A144299 J. Y. Choi and J. D. H. Smith, On the unimodilty and combinatorics of 
               Bessel numbers, Discrete Math., 264 (2003), 45-53.
%D A144299 E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.
%H A144299 David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">
               The Gift Exchange Problem</a> (arXiv:0907.0513, 2009)
%F A144299 T(n, k) = T(n-1, k-1) + (n-1)*T(n-2, k-1).
%F A144299 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.)
%e A144299 Triangle begins:
%e A144299 n:
%e A144299 0: 1
%e A144299 1: 1 0
%e A144299 2: 1 1 0
%e A144299 3: 1 3 0 0
%e A144299 4: 1 6 3 0 0
%e A144299 5: 1 10 15 0 0 0
%e A144299 6: 1 15 45 15 0 0 0
%e A144299 7: 1 21 105 105 0 0 0 0
%e A144299 8: 1 28 210 420 105 0 0 0 0
%e A144299 9: 1 36 378 1260 945 0 0 0 0 0
%e A144299 ...
%e A144299 The row sums give A000085.
%e A144299 For some purposes it is convenient to rotate the triangle by 45 degrees:
%e A144299 1.0.0.0.0..0..0...0...0....0....0.....0....
%e A144299 ..1.1.0.0..0..0...0...0....0....0.....0....
%e A144299 ....1.3.3..0..0...0...0....0....0.....0....
%e A144299 ......1.6.15.15...0...0....0....0.....0....
%e A144299 ........1.10.45.105.105....0....0.....0....
%e A144299 ...........1.15.105.420..945..945.....0....
%e A144299 ..............1..21.210.1260.4725.10395....
%e A144299 ..................1..28..378.3150.17325....
%e A144299 ......................1...36..630..6930....
%e A144299 ...........................1...45...990....
%e A144299 ...
%e A144299 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.
%e A144299 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).
%e A144299 Then b1(n,k) is the number of partitions of [1, 2, ...,k] into exactly 
               n blocks, each of size 1 or 2.
%p A144299 Maple code producing the rotated version:
%p A144299 b1 := proc(n, k)
%p A144299 option remember;
%p A144299 if n = k then 1;
%p A144299 elif k < n then 0;
%p A144299 elif n < 1 then 0;
%p A144299 else b1(n - 1, k - 1) + (k - 1)*b1(n - 1, k - 2);
%p A144299 end if;
%p A144299 end proc;
%p A144299 for n from 0 to 12 do lprint([seq(b1(n,k),k=0..2*n)]); od:
%t A144299 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 *)
%Y A144299 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.
%Y A144299 Cf. A144385, A144643.
%Y A144299 Cf. A118930 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 08 2008]
%Y A144299 Sequence in context: A126723 A090030 A080159 this_sequence A060514 A096936 
               A115979
%Y A144299 Adjacent sequences: A144296 A144297 A144298 this_sequence A144300 A144301 
               A144302
%K A144299 nonn,tabl,easy
%O A144299 1,8
%A A144299 David Applegate and N. J. A. Sloane (njas(AT)research.att.com), Dec 06 
               2008

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