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A100862 Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'. 5
1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 55, 55, 15, 1, 1, 21, 120, 215, 120, 21, 1, 1, 28, 231, 665, 665, 231, 28, 1, 1, 36, 406, 1736, 2835, 1736, 406, 36, 1, 1, 45, 666, 3990, 9891, 9891, 3990, 666, 45, 1, 1, 55, 1035, 8310, 29505, 45297, 29505, 8310, 1035, 55, 1, 1, 66, 1540, 16005, 77715, 173712, 173712, 77715, 16005, 1540, 66, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row n has n+1 terms.

Row sums yield A005425.

T(n,k) = T(n,n-k).

LINKS

Table of n, a(n) for n=0..77.

P. Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5

P. Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.

Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.

FORMULA

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

Row generating polynomial=P[n]=[ -i*sqrt(t/2)]^n*H(n, i(1+t)/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]=(1+t)P[n-1]+(n-1)tP[n-2].

EXAMPLE

T(3,2)=6 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following six 2-matchings: {Aa,BC},{Bb,AC},{Cc,AB},{Aa,Bb},{Aa,Cc} and {Bb,Cc}.

Triangle starts:

1;

1,1;

1,3,1;

1,6,6,1;

1,10,21,10,1;

1,15,55,55,15,1;

MAPLE

P[0]:=1: for n from 1 to 11 do P[n]:=sort(expand((1+t)*P[n-1]+(n-1)*t*P[n-2])) od: for n from 0 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # gives the sequence in triangular form

MATHEMATICA

P[0] = 1; P[1] = 1+t; P[n_] := P[n] = (1+t) P[n-1] + (n-1) t P[n-2];

Table[CoefficientList[P[n], t], {n, 0, 11}] // Flatten (* Jean-Fran├žois Alcover, Jul 23 2018 *)

PROG

(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); n!*polcoeff(polcoeff(exp(X+Y*X^2/2+X*Y), n, x), k, y)} \\ Paul D. Hanna

CROSSREFS

Cf. A005425.

Cf. A107102 (matrix inverse).

Sequence in context: A162747 A107105 A088925 * A098568 A180959 A131235

Adjacent sequences:  A100859 A100860 A100861 * A100863 A100864 A100865

KEYWORD

nonn,tabl

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.)