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

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, 36, 378, 1260, 945, 1, 45, 630, 3150, 4725, 945, 1, 55, 990, 6930, 17325, 10395, 1, 66, 1485, 13860, 51975, 62370, 10395, 1, 78, 2145, 25740, 135135, 270270, 135135, 1, 91, 3003, 45045, 315315, 945945, 945945, 135135

Offset: 0

Author

Emeric Deutsch, Jan 08 2005

Keywords

• nonn
• tabf
• nice

Comments

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

Inverse binomial transform is triangle with T(2n,n) = (2n-1)!!, 0 otherwise. - Paul Barry, May 21 2005

Equivalently, number of involutions of n with k pairs. - Franklin T. Adams-Watters, Jun 09 2006

From Gary W. Adamson, Dec 09 2009: (Start)

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

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

(End)

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

Row n is the list of coefficients of the independence polynomial of the n-triangular graph. - Eric W. Weisstein, Nov 11 2016

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

Examples

			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}.
Triangle starts:
[0] 1;
[1] 1;
[2] 1,  1;
[3] 1,  3;
[4] 1,  6,   3;
[5] 1, 10,  15;
[6] 1, 15,  45,   15;
[7] 1, 21, 105,  105;
[8] 1, 28, 210,  420, 105;
[9] 1, 36, 378, 1260, 945.
.
From _Eric W. Weisstein_, Nov 11 2016: (Start)
As polynomials:
1,
1,
1 + x,
1 + 3*x,
1 + 6*x + 3*x^2,
1 + 10*x + 15*x^2,
1 + 15*x + 45*x^2 + 15*x^3. (End)
		

References

• M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (1983 reprint), 10th edition, 1964, expression 22.3.11 in page 775.
• C. D. Godsil, Algebraic Combinatorics, Chapman & Hall, New York, 1993.

Links

• T. D. Noe, Rows n = 0..100, flattened
• Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
• Gérard Le Caër, A new family of solvable Pearson-Dirichlet random walks, Journal of Statistical Physics 144:1 (2011), pp. 23-45.
• Ji Young Choi and Jonathan D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.
• Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
• John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016.
• Mikael Fremling, On the modular covariance properties of composite fermions on the torus, arXiv:1810.10391 [cond-mat.str-el], 2018.
• Gary R. W. Greaves, Jeven Syatriadi, and Charissa I. Utomo, Chromatic polynomials of signed graphs and dominating-vertex deletion formulae, arXiv:2407.00883 [math.CO], 2024. See p. 11.
• A. Hernando, R. Hernando, A. Plastino and A. R. Plastino, The workings of the Maximum Entropy Principle in collective human behavior, arXiv preprint arXiv:1201.0905 [stat.AP], 2012.
• Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
• Aaron Pollack, Exceptional theta functions, arXiv:2211.05280 [math.NT], Nov 2022. See Lemma 7.5.1.
• Eric Weisstein's World of Mathematics, Complete Graph
• Eric Weisstein's World of Mathematics, Independence Polynomial
• Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
• Eric Weisstein's World of Mathematics, Triangular Graph
• Wikipedia, Normal distribution, section 'Moments'
• Jian Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Crossrefs

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

Cf. A000085 (row sums).

Cf. A118930, A008619.

Programs

Haskell

a100861 n k = a100861_tabf !! n !! k
a100861_row n = a100861_tabf !! n
a100861_tabf = zipWith take a008619_list a144299_tabl
-- Reinhard Zumkeller, Jan 02 2014

Maple

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
# Alternative:
A100861 := proc(n,k)
    n!/k!/(n-2*k)!/2^k ;
end proc:
seq(seq(A100861(n,k),k=0..n/2),n=0..10) ; # R. J. Mathar, Aug 19 2014

Mathematica

Table[Table[n!/(i! 2^i (n - 2 i)!), {i, 0, Floor[n/2]}], {n, 0, 10}] // Flatten  (* Geoffrey Critzer, Mar 27 2011 *)
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 *)
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 *)

PARI

T(n,k)=if(k<0 || 2*k>n, 0, n!/k!/(n-2*k)!/2^k) /* Michael Somos, Jun 04 2005 */

Formula

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

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

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

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].

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

T(n,k) = (n-2k+1)*T(n-1,k-1) + T(n-1,k). - Franklin T. Adams-Watters, Jun 09 2006

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

T(n,k) = A144299(n,k), k=0..n/2. - Reinhard Zumkeller, Jan 02 2014