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A100861 Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).

%I A100861 #119 Feb 16 2025 08:32:55
%S A100861 1,1,1,1,1,3,1,6,3,1,10,15,1,15,45,15,1,21,105,105,1,28,210,420,105,1,
%T A100861 36,378,1260,945,1,45,630,3150,4725,945,1,55,990,6930,17325,10395,1,
%U A100861 66,1485,13860,51975,62370,10395,1,78,2145,25740,135135,270270,135135,1,91,3003,45045,315315,945945,945945,135135
%N A100861 Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
%C A100861 Row n contains 1 + floor(n/2) terms. Row sums yield A000085. T(2n,n) = T(2n-1,n-1) = (2n-1)!! (A001147).
%C A100861 Inverse binomial transform is triangle with T(2n,n) = (2n-1)!!, 0 otherwise. - _Paul Barry_, May 21 2005
%C A100861 Equivalently, number of involutions of n with k pairs. - _Franklin T. Adams-Watters_, Jun 09 2006
%C A100861 From _Gary W. Adamson_, Dec 09 2009: (Start)
%C A100861 If considered as an infinite lower triangular matrix (cf. A144299),
%C A100861 lim_{n->} A100861^n = A118930: (1, 1, 2, 4, 13, 41, ...).
%C A100861 (End)
%C A100861 Sum_{k=0..floor(n/2)} T(n,k)m^(n-2k)s^(2k) is the n-th non-central moment of the normal probability distribution with mean m and standard deviation s. - _Stanislav Sykora_, Jun 19 2014
%C A100861 Row n is the list of coefficients of the independence polynomial of the n-triangular graph. - _Eric W. Weisstein_, Nov 11 2016
%C A100861 Restating the 2nd part of the Name, row n is the list of coefficients of the matching-generating polynomial of the complete graph K_n. - _Eric W. Weisstein_, Apr 03 2018
%D A100861 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (1983 reprint), 10th edition, 1964, expression 22.3.11 in page 775.
%D A100861 C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.
%H A100861 T. D. Noe, <a href="/A100861/b100861.txt">Rows n = 0..100, flattened</a>
%H A100861 Paul Barry, <a href="https://arxiv.org/abs/1804.05027">The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1804.05027 [math.CO], 2018.
%H A100861 Gérard Le Caër, <a href="http://dx.doi.org/10.1007/s10955-011-0245-4">A new family of solvable Pearson-Dirichlet random walks</a>, Journal of Statistical Physics 144:1 (2011), pp. 23-45.
%H A100861 Ji Young Choi and Jonathan D. H. Smith, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00549-6">On the unimodality and combinatorics of Bessel numbers</a>, Discrete Math., 264 (2003), 45-53.
%H A100861 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>, 2012.
%H A100861 John Engbers, David Galvin, and Clifford Smyth, <a href="https://arxiv.org/abs/1610.05803">Restricted Stirling and Lah numbers and their inverses</a>, arXiv:1610.05803 [math.CO], 2016.
%H A100861 Mikael Fremling, <a href="https://arxiv.org/1810.10391">On the modular covariance properties of composite fermions on the torus</a>, arXiv:1810.10391 [cond-mat.str-el], 2018.
%H A100861 Gary R. W. Greaves, Jeven Syatriadi, and Charissa I. Utomo, <a href="https://arxiv.org/abs/2407.00883">Chromatic polynomials of signed graphs and dominating-vertex deletion formulae</a>, arXiv:2407.00883 [math.CO], 2024. See p. 11.
%H A100861 A. Hernando, R. Hernando, A. Plastino and A. R. Plastino, <a href="http://arxiv.org/abs/1201.0905">The workings of the Maximum Entropy Principle in collective human behavior</a>, arXiv preprint arXiv:1201.0905 [stat.AP], 2012.
%H A100861 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 A100861 Aaron Pollack, <a href="https://arxiv.org/abs/2211.05280">Exceptional theta functions</a>, arXiv:2211.05280 [math.NT], Nov 2022. See Lemma 7.5.1.
%H A100861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>
%H A100861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependencePolynomial.html">Independence Polynomial</a>
%H A100861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching-GeneratingPolynomial.html">Matching-Generating Polynomial</a>
%H A100861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>
%H A100861 Wikipedia, <a href="http://en.wikipedia.org/wiki/Normal_distribution">Normal distribution</a>, section 'Moments'
%H A100861 Jian Zhou, <a href="http://arxiv.org/abs/1405.5296">Quantum deformation theory of the Airy curve and the mirror symmetry of a point</a>, arXiv preprint arXiv:1405.5296 [math.AG], 2014.
%F A100861 T(n, k) = n!/(k!(n-2k)!*2^k).
%F A100861 E.g.f.: exp(z+tz^2/2).
%F A100861 G.f.: g(t, z) satisfies the differential equation g = 1 + zg + tz^2*(d/dz)(zg).
%F A100861 Row generating polynomial = P[n] = [-i*sqrt(t/2)]^n*H(n, i/sqrt(2t)), where H(n, x) is a Hermite polynomial and i=sqrt(-1). Row generating polynomials P[n] satisfy P[0]=1, P[n] = P[n-1] + (n-1)tP[n-2].
%F A100861 T(n, k) = binomial(n, 2k)(2k-1)!!. - _Paul Barry_, May 21 2002 [Corrected by Roland Hildebrand, Mar 06 2009]
%F A100861 T(n,k) = (n-2k+1)*T(n-1,k-1) + T(n-1,k). - _Franklin T. Adams-Watters_, Jun 09 2006
%F A100861 E.g.f.: 1 + (x+y*x^2/2)/(E(0)-(x+y*x^2/2)), where E(k) = 1 + (x+y*x^2/2)/(1 + (k+1)/E(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 08 2013
%F A100861 T(n,k) = A144299(n,k), k=0..n/2. - _Reinhard Zumkeller_, Jan 02 2014
%e A100861 T(4, 2) = 3 because in the graph with vertex set {A, B, C, D} and edge set {AB, BC, CD, AD, AC, BD} we have the following three 2-matchings: {AB, CD},{AC, BD} and {AD, BC}.
%e A100861 Triangle starts:
%e A100861 [0] 1;
%e A100861 [1] 1;
%e A100861 [2] 1,  1;
%e A100861 [3] 1,  3;
%e A100861 [4] 1,  6,   3;
%e A100861 [5] 1, 10,  15;
%e A100861 [6] 1, 15,  45,   15;
%e A100861 [7] 1, 21, 105,  105;
%e A100861 [8] 1, 28, 210,  420, 105;
%e A100861 [9] 1, 36, 378, 1260, 945.
%e A100861 .
%e A100861 From _Eric W. Weisstein_, Nov 11 2016: (Start)
%e A100861 As polynomials:
%e A100861 1,
%e A100861 1,
%e A100861 1 + x,
%e A100861 1 + 3*x,
%e A100861 1 + 6*x + 3*x^2,
%e A100861 1 + 10*x + 15*x^2,
%e A100861 1 + 15*x + 45*x^2 + 15*x^3. (End)
%p A100861 P[0]:=1: for n from 1 to 14 do P[n]:=sort(expand(P[n-1]+(n-1)*t*P[n-2])) od: for n from 0 to 14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields the sequence in triangular form
%p A100861 # Alternative:
%p A100861 A100861 := proc(n,k)
%p A100861     n!/k!/(n-2*k)!/2^k ;
%p A100861 end proc:
%p A100861 seq(seq(A100861(n,k),k=0..n/2),n=0..10) ; # _R. J. Mathar_, Aug 19 2014
%t A100861 Table[Table[n!/(i! 2^i (n - 2 i)!), {i, 0, Floor[n/2]}], {n, 0, 10}] // Flatten  (* _Geoffrey Critzer_, Mar 27 2011 *)
%t A100861 CoefficientList[Table[2^(n/2) (-(1/x))^(-n/2) HypergeometricU[-n/2, 1/2, -1/(2 x)], {n, 0, 10}], x] // Flatten (* _Eric W. Weisstein_, Apr 03 2018 *)
%t A100861 CoefficientList[Table[(-I)^n Sqrt[x/2]^n HermiteH[n, I/Sqrt[2 x]], {n, 0, 10}], x] // Flatten (* _Eric W. Weisstein_, Apr 03 2018 *)
%o A100861 (PARI) T(n,k)=if(k<0 || 2*k>n, 0, n!/k!/(n-2*k)!/2^k) /* _Michael Somos_, Jun 04 2005 */
%o A100861 (Haskell)
%o A100861 a100861 n k = a100861_tabf !! n !! k
%o A100861 a100861_row n = a100861_tabf !! n
%o A100861 a100861_tabf = zipWith take a008619_list a144299_tabl
%o A100861 -- _Reinhard Zumkeller_, Jan 02 2014
%Y A100861 Other versions of this same triangle are given in A144299, A001497, A001498, A111924.
%Y A100861 Cf. A000085 (row sums).
%Y A100861 Cf. A118930, A008619.
%K A100861 nonn,tabf,nice
%O A100861 0,6
%A A100861 _Emeric Deutsch_, Jan 08 2005