A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

1, 1, 1, 1, 4, 3, 1, 7, 15, 9, 1, 10, 36, 54, 27, 1, 13, 66, 162, 189, 81, 1, 16, 105, 360, 675, 648, 243, 1, 19, 153, 675, 1755, 2673, 2187, 729, 1, 22, 210, 1134, 3780, 7938, 10206, 7290, 2187, 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561, 1, 28, 351, 2592, 12474, 40824, 91854, 139968, 137781, 78732, 19683

Offset: 0

Author

N. J. A. Sloane, May 03 2000

Keywords

• tabl
• nonn
• easy

Comments

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

Triangle read by rows, n-th row = X^(n-1) * [1, 1, 0, 0, 0, ...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (3,3,3,...) in the subdiagonal; given row 0 = 1. - Gary W. Adamson, Jul 19 2008

Fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(2x+1)^n; see A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. - Clark Kimberling, Aug 04 2011

Examples

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  7,  15,   9;
  1, 10,  36,  54,   27;
  1, 13,  66, 162,  189,   81;
  1, 16, 105, 360,  675,  648,  243;
  1, 19, 153, 675, 1755, 2673, 2187, 729;
		

Links

• G. C. Greubel, Rows n = 0..50 of the 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.

Crossrefs

Cf. A000007, A000012, A001296, A006138, A006234, A016777, A024462.

Cf. A051836, A062741, A080420, A080421, A080422, A080423, A081294.

Cf. A084938, A098399, A110523, A133494, A136158, A193722.

Programs

Magma

A038763:= func< n,k | n eq 0 select 1 else 3^(k-1)*(3*n-2*k)*Binomial(n,k)/n >;
[A038763(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 27 2023

Mathematica

A038763[n_,k_]:= If[n==0, 1, 3^(k-1)*(3*n-2*k)*Binomial[n,k]/n];
Table[A038763[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 27 2023 *)

PARI

T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); T(n-1,k) + 3*T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jul 25 2023

SageMath

def A038763(n,k): return 1 if (n==0) else 3^(k-1)*(3*n-2*k)*binomial(n,k)/n
flatten([[A038763(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 27 2023

Formula

T(n, 0)=1; T(1, 1)=1; T(n, k)=0 for k>n; T(n, k) = T(n-1, k-1)*3 + T(n-1, k) for n >= 2.

Sum_{k=0..n} T(n,k) = A081294(n). - Philippe Deléham, Sep 22 2006

T(n, k) = A136158(n, n-k). - Philippe Deléham, Dec 17 2007

G.f.: (1-2*x*y)/(1-(3*y+1)*x). - R. J. Mathar, Aug 11 2015

From G. C. Greubel, Dec 27 2023: (Start)

T(n, 0) = A000012(n).

T(n, 1) = A016777(n-1).

T(n, 2) = A062741(n-1).

T(n, 3) = 9*A002411(n-2).

T(n, 4) = 27*A001296(n-3).

T(n, 5) = 81*A051836(n-4).

T(n, n) = A133494(n).

T(n, n-1) = A006234(n+2).

T(n, n-2) = A080420(n-2).

T(n, n-3) = A080421(n-3).

T(n, n-4) = A080422(n-4).

T(n, n-5) = A080423(n-5).

T(2*n, n) = 4*A098399(n-1) + (2/3)*[n=0].

Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A006138(n-1) + (2/3)*[n=0].

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A110523(n-1) + (4/3)*[n=0]. (End)

Extensions

More terms from Michel Marcus, Jul 25 2023