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

%I A038763 #37 Dec 30 2023 23:50:11
%S A038763 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,
%T A038763 675,648,243,1,19,153,675,1755,2673,2187,729,1,22,210,1134,3780,7938,
%U A038763 10206,7290,2187,1,25,276,1764,7182,19278,34020,37908,24057,6561,1,28,351,2592,12474,40824,91854,139968,137781,78732,19683
%N A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.
%C A038763 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
%C A038763 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
%C A038763 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
%H A038763 G. C. Greubel, <a href="/A038763/b038763.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A038763 S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, <a href="https://hrcak.srce.hr/177109">Unbranched catacondensed polygonal systems containing hexagons and tetragons</a>, Croatica Chem. Acta, 69 (1996), 757-774.
%F A038763 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.
%F A038763 Sum_{k=0..n} T(n,k) = A081294(n). - _Philippe Deléham_, Sep 22 2006
%F A038763 T(n, k) = A136158(n, n-k). - _Philippe Deléham_, Dec 17 2007
%F A038763 G.f.: (1-2*x*y)/(1-(3*y+1)*x). - _R. J. Mathar_, Aug 11 2015
%F A038763 From _G. C. Greubel_, Dec 27 2023: (Start)
%F A038763 T(n, 0) = A000012(n).
%F A038763 T(n, 1) = A016777(n-1).
%F A038763 T(n, 2) = A062741(n-1).
%F A038763 T(n, 3) = 9*A002411(n-2).
%F A038763 T(n, 4) = 27*A001296(n-3).
%F A038763 T(n, 5) = 81*A051836(n-4).
%F A038763 T(n, n) = A133494(n).
%F A038763 T(n, n-1) = A006234(n+2).
%F A038763 T(n, n-2) = A080420(n-2).
%F A038763 T(n, n-3) = A080421(n-3).
%F A038763 T(n, n-4) = A080422(n-4).
%F A038763 T(n, n-5) = A080423(n-5).
%F A038763 T(2*n, n) = 4*A098399(n-1) + (2/3)*[n=0].
%F A038763 Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
%F A038763 Sum_{k=0..floor(n/2)} T(n-k, k) = A006138(n-1) + (2/3)*[n=0].
%F A038763 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A110523(n-1) + (4/3)*[n=0]. (End)
%e A038763 Triangle begins:
%e A038763   1;
%e A038763   1,  1;
%e A038763   1,  4,   3;
%e A038763   1,  7,  15,   9;
%e A038763   1, 10,  36,  54,   27;
%e A038763   1, 13,  66, 162,  189,   81;
%e A038763   1, 16, 105, 360,  675,  648,  243;
%e A038763   1, 19, 153, 675, 1755, 2673, 2187, 729;
%t A038763 A038763[n_,k_]:= If[n==0, 1, 3^(k-1)*(3*n-2*k)*Binomial[n,k]/n];
%t A038763 Table[A038763[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 27 2023 *)
%o A038763 (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);
%o A038763 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ _Michel Marcus_, Jul 25 2023
%o A038763 (Magma)
%o A038763 A038763:= func< n,k | n eq 0 select 1 else 3^(k-1)*(3*n-2*k)*Binomial(n,k)/n >;
%o A038763 [A038763(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 27 2023
%o A038763 (SageMath)
%o A038763 def A038763(n,k): return 1 if (n==0) else 3^(k-1)*(3*n-2*k)*binomial(n,k)/n
%o A038763 flatten([[A038763(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Dec 27 2023
%Y A038763 Cf. A000007, A000012, A001296, A006138, A006234, A016777, A024462.
%Y A038763 Cf. A051836, A062741, A080420, A080421, A080422, A080423, A081294.
%Y A038763 Cf. A084938, A098399, A110523, A133494, A136158, A193722.
%K A038763 tabl,nonn,easy
%O A038763 0,5
%A A038763 _N. J. A. Sloane_, May 03 2000
%E A038763 More terms from _Michel Marcus_, Jul 25 2023