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

0,5

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

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.

Sequence in context: A090210 A248925 A168131 * A049252 A098315 A006704

Adjacent sequences: A024459 A024460 A024461 * A024463 A024464 A024465

Keyword

tabl,nonn,easy

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