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%I #90 Aug 23 2023 10:45:33
%S 1,0,1,0,1,1,0,0,3,1,0,0,3,6,1,0,0,0,15,10,1,0,0,0,15,45,15,1,0,0,0,0,
%T 105,105,21,1,0,0,0,0,105,420,210,28,1,0,0,0,0,0,945,1260,378,36,1,0,
%U 0,0,0,0,945,4725,3150,630,45,1,0,0,0,0,0,0,10395,17325,6930,990,55,1,0,0
%N Exponential Riordan array (1, x(1+x/2)).
%C Entries are Bessel polynomial coefficients. Row sums are A000085. Diagonal sums are A122849. Inverse is A122850. Product of A007318 and A122848 gives A100862.
%C T(n,k) is the number of self-inverse permutations of {1,2,...,n} having exactly k cycles. - _Geoffrey Critzer_, May 08 2012
%C Bessel numbers of the second kind. For relations to the Hermite polynomials and the Catalan (A033184 and A009766) and Fibonacci (A011973, A098925, and A092865) matrices, see Yang and Qiao. - _Tom Copeland_, Dec 18 2013.
%C Also the inverse Bell transform of the double factorial of odd numbers Product_{k= 0..n-1} (2*k+1) (A001147). For the definition of the Bell transform see A264428 and for cross-references A265604. - _Peter Luschny_, Dec 31 2015
%H G. C. Greubel, <a href="/A122848/b122848.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Peter Bala, <a href="/A035342/a035342_Bala.txt">Generalized Dobinski formulas</a>
%H Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Celeste/celeste3.html"> Two Approaches to Normal Order Coefficients</a>. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>
%H H. Han and S. Seo, <a href="http://dx.doi.org/10.1016/j.ejc.2007.12.002">Combinatorial proofs of inverse relations and log-concavity for Bessel numbers</a>, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
%H Robert S. Maier, <a href="https://arxiv.org/abs/2308.10332">Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers</a>, arXiv:2308.10332 [math.CO], 2023. See p. 18.
%H S. Yang and Z. Qiao, <a href="http://jmre.ijournals.cn/en/ch/reader/create_pdf.aspx?file_no=20110407&year_id=2011&quarter_id=4&falg=1">The Bessel Numbers and Bessel Matrices</a>, Journal of Mathematical Research & Exposition, July, 2011, Vol. 31, No. 4, pp. 627-636. [From _Tom Copeland_, Dec 18 2013]
%F Number triangle T(n,k) = k!*C(n,k)/((2k-n)!*2^(n-k)).
%F T(n,k) = A001498(k,n-k). - _Michael Somos_, Oct 03 2006
%F E.g.f.: exp(y(x+x^2/2)). - _Geoffrey Critzer_, May 08 2012
%F Triangle equals the matrix product A008275*A039755. Equivalently, the n-th row polynomial R(n,x) is given by the Type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} P(n,2*k+1)*(x/2)^k/k!, where P(n,x) = x*(x-1)*...*(x-n+1) denotes the falling factorial polynomial. Cf. A113278. - _Peter Bala_, Jun 23 2014
%F From _Daniel Checa_, Aug 28 2022: (Start)
%F E.g.f. for the m-th column: (x^2/2+x)^m/m!.
%F T(n,k) = T(n-1,k-1) + (n-1)*T(n-2,k-1) for n>1 and k=1..n, T(0,0) = 1. (End)
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 0 3 1
%e 0 0 3 6 1
%e 0 0 0 15 10 1
%e 0 0 0 15 45 15 1
%e 0 0 0 0 105 105 21 1
%e 0 0 0 0 105 420 210 28 1
%e 0 0 0 0 0 945 1260 378 36 1
%e From _Gus Wiseman_, Jan 12 2021: (Start)
%e As noted above, a(n) is the number of set partitions of {1..n} into k singletons or pairs. This is also the number of set partitions of subsets of {1..n} into n - k pairs. In the first case, row n = 5 counts the following set partitions:
%e {{1},{2,3},{4,5}} {{1},{2},{3},{4,5}} {{1},{2},{3},{4},{5}}
%e {{1,2},{3},{4,5}} {{1},{2},{3,4},{5}}
%e {{1,2},{3,4},{5}} {{1},{2,3},{4},{5}}
%e {{1,2},{3,5},{4}} {{1,2},{3},{4},{5}}
%e {{1},{2,4},{3,5}} {{1},{2},{3,5},{4}}
%e {{1},{2,5},{3,4}} {{1},{2,4},{3},{5}}
%e {{1,3},{2},{4,5}} {{1},{2,5},{3},{4}}
%e {{1,3},{2,4},{5}} {{1,3},{2},{4},{5}}
%e {{1,3},{2,5},{4}} {{1,4},{2},{3},{5}}
%e {{1,4},{2},{3,5}} {{1,5},{2},{3},{4}}
%e {{1,4},{2,3},{5}}
%e {{1,4},{2,5},{3}}
%e {{1,5},{2},{3,4}}
%e {{1,5},{2,3},{4}}
%e {{1,5},{2,4},{3}}
%e In the second case, we have:
%e {{1,2},{3,4}} {{1,2}} {}
%e {{1,2},{3,5}} {{1,3}}
%e {{1,2},{4,5}} {{1,4}}
%e {{1,3},{2,4}} {{1,5}}
%e {{1,3},{2,5}} {{2,3}}
%e {{1,3},{4,5}} {{2,4}}
%e {{1,4},{2,3}} {{2,5}}
%e {{1,4},{2,5}} {{3,4}}
%e {{1,4},{3,5}} {{3,5}}
%e {{1,5},{2,3}} {{4,5}}
%e {{1,5},{2,4}}
%e {{1,5},{3,4}}
%e {{2,3},{4,5}}
%e {{2,4},{3,5}}
%e {{2,5},{3,4}}
%e (End)
%p # The function BellMatrix is defined in A264428.
%p BellMatrix(n -> `if`(n<2,1,0), 9); # _Peter Luschny_, Jan 27 2016
%t t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
%t (* Second program: *)
%t rows = 12;
%t t = Join[{1, 1}, Table[0, rows]];
%t T[n_, k_] := BellY[n, k, t];
%t Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 23 2018,after _Peter Luschny_ *)
%t sbs[{}]:={{}};sbs[set:{i_,___}]:=Join@@Function[s,(Prepend[#1,s]&)/@sbs[Complement[set,s]]]/@Cases[Subsets[set],{i}|{i,_}];
%t Table[Length[Select[sbs[Range[n]],Length[#]==k&]],{n,0,6},{k,0,n}] (* _Gus Wiseman_, Jan 12 2021 *)
%o (PARI) {T(n,k)=if(2*k<n||k>n, 0, n!/(2*k-n)!/(n-k)!*2^(k-n))} /* _Michael Somos_, Oct 03 2006 */
%o (Sage) # uses[inverse_bell_transform from A265605]
%o multifact_2_1 = lambda n: prod(2*k + 1 for k in (0..n-1))
%o inverse_bell_matrix(multifact_2_1, 9) # _Peter Luschny_, Dec 31 2015
%Y Cf. A001497, A008275, A039755, A096713, A104556, A113278, A130757.
%Y Row sums are A000085.
%Y Column sums are A001515.
%Y Same as A049403 but with a first column k = 0.
%Y The same set partitions counted by number of pairs are A100861.
%Y Reversing rows gives A111924 (without column k = 0).
%Y A047884 counts standard Young tableaux by size and greatest row length.
%Y A238123 counts standard Young tableaux by size and least row length.
%Y A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
%Y A322661 counts labeled covering half-loop-graphs.
%Y A339742 counts factorizations into distinct primes or squarefree semiprimes.
%Y Cf. A000110, A000258, A320732, A321729, A339741, A339887.
%K easy,nonn,tabl
%O 0,9
%A _Paul Barry_, Sep 14 2006
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