A100861 - OEIS

%I #118 Jul 05 2024 10:31:07

%S 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 36,378,1260,945,1,45,630,3150,4725,945,1,55,990,6930,17325,10395,1,

%U 66,1485,13860,51975,62370,10395,1,78,2145,25740,135135,270270,135135,1,91,3003,45045,315315,945945,945945,135135

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

%C Row n contains 1 + floor(n/2) terms. Row sums yield A000085. T(2n,n) = T(2n-1,n-1) = (2n-1)!! (A001147).

%C Inverse binomial transform is triangle with T(2n,n) = (2n-1)!!, 0 otherwise. - _Paul Barry_, May 21 2005

%C Equivalently, number of involutions of n with k pairs. - _Franklin T. Adams-Watters_, Jun 09 2006

%C From _Gary W. Adamson_, Dec 09 2009: (Start)

%C If considered as an infinite lower triangular matrix (cf. A144299),

%C lim_{n->} A100861^n = A118930: (1, 1, 2, 4, 13, 41, ...).

%C (End)

%C 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 Row n is the list of coefficients of the independence polynomial of the n-triangular graph. - _Eric W. Weisstein_, Nov 11 2016

%C 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 M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (1983 reprint), 10th edition, 1964, expression 22.3.11 in page 775.

%D C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

%H T. D. Noe, <a href="/A100861/b100861.txt">Rows n = 0..100, flattened</a>

%H 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 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 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 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 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 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 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 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependencePolynomial.html">Independence Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching-GeneratingPolynomial.html">Matching-Generating Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Normal_distribution">Normal distribution</a>, section 'Moments'

%H 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 T(n, k) = n!/(k!(n-2k)!*2^k).

%F E.g.f.: exp(z+tz^2/2).

%F G.f.: g(t, z) satisfies the differential equation g = 1 + zg + tz^2*(d/dz)(zg).

%F 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 T(n, k) = binomial(n, 2k)(2k-1)!!. - _Paul Barry_, May 21 2002 [Corrected by Roland Hildebrand, Mar 06 2009]

%F T(n,k) = (n-2k+1)*T(n-1,k-1) + T(n-1,k). - _Franklin T. Adams-Watters_, Jun 09 2006

%F 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 T(n,k) = A144299(n,k), k=0..n/2. - _Reinhard Zumkeller_, Jan 02 2014

%e 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 Triangle starts:

%e [0] 1;

%e [1] 1;

%e [2] 1,  1;

%e [3] 1,  3;

%e [4] 1,  6,   3;

%e [5] 1, 10,  15;

%e [6] 1, 15,  45,   15;

%e [7] 1, 21, 105,  105;

%e [8] 1, 28, 210,  420, 105;

%e [9] 1, 36, 378, 1260, 945.

%e .

%e From _Eric W. Weisstein_, Nov 11 2016: (Start)

%e As polynomials:

%e 1,

%e 1,

%e 1 + x,

%e 1 + 3*x,

%e 1 + 6*x + 3*x^2,

%e 1 + 10*x + 15*x^2,

%e 1 + 15*x + 45*x^2 + 15*x^3. (End)

%p 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 # Alternative:

%p A100861 := proc(n,k)

%p     n!/k!/(n-2*k)!/2^k ;

%p end proc:

%p seq(seq(A100861(n,k),k=0..n/2),n=0..10) ; # _R. J. Mathar_, Aug 19 2014

%t Table[Table[n!/(i! 2^i (n - 2 i)!), {i, 0, Floor[n/2]}], {n, 0, 10}] // Flatten  (* _Geoffrey Critzer_, Mar 27 2011 *)

%t 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 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 (PARI) T(n,k)=if(k<0 || 2*k>n, 0, n!/k!/(n-2*k)!/2^k) /* _Michael Somos_, Jun 04 2005 */

%o (Haskell)

%o a100861 n k = a100861_tabf !! n !! k

%o a100861_row n = a100861_tabf !! n

%o a100861_tabf = zipWith take a008619_list a144299_tabl

%o -- _Reinhard Zumkeller_, Jan 02 2014

%Y Other versions of this same triangle are given in A144299, A001497, A001498, A111924.

%Y Cf. A000085 (row sums).

%Y Cf. A118930, A008619.

%K nonn,tabf,nice

%O 0,6

%A _Emeric Deutsch_, Jan 08 2005