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[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)
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-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
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Jan 08 2005
STATUS
approved