A024462 Triangle T(n,k) read by rows, arising in enumeration of catafusenes.

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, 1, 23, 232, 1344, 4914, 11718, 18144, 17496, 9477, 2187, 1, 26, 301, 2040, 8946, 26460, 53298, 71928, 61965, 30618, 6561

Offset: 0

Author

N. J. A. Sloane, May 03 2000

Keywords

• tabl
• nonn
• easy

Examples

			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;
   ...
		

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).

Crossrefs

Cf. A038763.

Left-edge columns (essentially) include A016789 and A038764. Right-edge diagonal columns (essentially) include A000244, A038765, and A081892. Row sums are (essentially) A000302.

Programs

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 *)

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

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)

Extensions

More terms from James Sellers, May 03 2000

Edited by Ralf Stephan, Jan 25 2005