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'.
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
Offset: 0
Examples
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;
Links
- Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq., Vol. 14 (2011), Article 11.4.5.
- Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, Vol. 17 (2014), Article 14.2.6.
- Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
Programs
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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
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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-François Alcover, Jul 23 2018 *) -
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, Jul 18 2005
Formula
T(n,k) = T(n,n-k).
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].
From Fabián Pereyra, Jan 05 2022: (Start)
T(n,k) = T(n-1,k) + (n-1)*T(n-2,k-1) + T(n-1,k-1), n>=0, 0<=k<=n; T(0,0) = 1.
T(n,k) = Sum_{j=k..n} C(n,j)*B(j,j-k), where B are the Bessel numbers A100861. (End)
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