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

0,5

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

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.

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)

Example

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;

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 *)

Prog

(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
(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
(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

Crossrefs

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

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

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

Sequence in context: A016698 A200115 A332496 * A200384 A128007 A098458

Adjacent sequences: A038760 A038761 A038762 * A038764 A038765 A038766

Keyword

tabl,nonn,easy

Author

N. J. A. Sloane, May 03 2000

Extensions

More terms from Michel Marcus, Jul 25 2023

Status

approved