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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. 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. 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=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) 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:=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 = 1; P = 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 *) 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, Jul 18 2005 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 August 11 15:10 EDT 2022. Contains 356066 sequences. (Running on oeis4.)