OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..20 of triangle, flattened
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774; see Table III (p. 767).
FORMULA
T(n, k) = 3 * T(n-1, k-1) + T(n-1, k), starting with [1], [1, 1], [1, 2, 1].
From Petros Hadjicostas, May 27 2019: (Start)
T(n, k) = (n-2)!/(k! * (n-k)!) * (9*n*(n-1) - 4*k*(3*n-k-2)) * 3^(k-2) for n >= max(k, 2) and k >= 0. (See the top formula of p. 767 in Cyvin et al. (1996).)
Bivariate g.f.: Sum_{n, k >= 0} T(n, k) * x^n * y^k = 1 + x * (1 + y) + x^2 * (1 + y)^2/(1 - x - 3 * x * y).
(End)
EXAMPLE
Triangle begins (rows indexed by n >= 0 and columns by k >= 0):
1;
1, 1;
1, 2, 1;
1, 5, 7, 3;
1, 8, 22, 24, 9;
1, 11, 46, 90, 81, 27;
1, 14, 79, 228, 351, 270, 81;
1, 17, 121, 465, 1035, 1323, 891, 243;
1, 20, 172, 828, 2430, 4428, 4860, 2916, 729;
...
MAPLE
## The following Maple program gives the Taylor expansion of the bivariate g.f. of T(n, k) in powers of x:
T := proc (x, y) 1+x*(y+1)+x^2*(y+1)^2/(1-x-3*y*x) end proc;
expand(taylor(T(x, y), x = 0, 20)); ## Petros Hadjicostas, May 27 2019
MATHEMATICA
T[n_, 0]:= 1; T[n_, k_]:= If[k<0 || k>n, 0, If[n==1 && k==1, 1, If[n==2 && k==1, 2, If[k==n && n>=2, 3^(n-2), 3*T[n-1, k-1] + T[n-1, k]]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, May 30 2019 *)
PROG
(PARI) T(n, k)=if(n<0||k<0||k>n, 0, if(n<3, [[1], [1, 1], [1, 2, 1]][n+1][k+1], 3*T(n-1, k-1)+T(n-1, k))) \\ Ralf Stephan, Jan 25 2005
(Sage)
def T(n, k):
if (k<0 and k>n): return 0
elif (k==0): return 1
elif (n==k==1): return 1
elif (n==2 and k==1): return 2
elif (n>=2 and k==n): return 3^(n-2)
else: return 3*T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 30 2019
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, May 03 2000
EXTENSIONS
More terms from James A. Sellers, May 03 2000
Edited by Ralf Stephan, Jan 25 2005
STATUS
approved