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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). 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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

REFERENCES

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.

G. Le Caer, A new family of solvable Pearson-Dirichlet random walks, Journal of Statistical Physics 144:1 (2011), pp. 23-45.

J. Y. Choi and J. D. H. Smith, On the unimodality and combinatorics of Bessel numbers, Discrete Math., 264 (2003), 45-53.

T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.

John Engbers, David Galvin, 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.

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.

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'

J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

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

EXAMPLE

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:

1,

1,

1,  1,

1,  3,

1,  6,  3,

1, 10, 15,

1, 15, 45, 15,

...

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)

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 *)

PROG

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

(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

CROSSREFS

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

Cf. A000085 (row sums).

Cf. A118930, A008619.

Sequence in context: A131110 A133093 A065567 * A131031 A130452 A133085

Adjacent sequences:  A100858 A100859 A100860 * A100862 A100863 A100864

KEYWORD

nonn,tabf,nice

AUTHOR

Emeric Deutsch, Jan 08 2005

STATUS

approved

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Last modified September 28 17:43 EDT 2020. Contains 337393 sequences. (Running on oeis4.)