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A323943 Trapezoidal matrix T(n,k) (n>=1, 1<=k<=n+2) read by rows, arising in enumeration of unbranched k-4-catafusenes.

%I A323943 #18 Nov 08 2023 12:35:04
%S A323943 1,2,1,3,7,5,1,9,24,22,8,1,27,81,90,46,11,1,81,270,351,228,79,14,1,
%T A323943 243,891,1323,1035,465,121,17,1,729,2916,4860,4428,2430,828,172,20,1,
%U A323943 2187,9477,17496,18144,11718,4914,1344,232,23,1,6561,30618,61965,71928,53298,26460,8946,2040,301,26,1,19683
%N A323943 Trapezoidal matrix T(n,k) (n>=1, 1<=k<=n+2) read by rows, arising in enumeration of unbranched k-4-catafusenes.
%C A323943 Rows sums are powers of 4.
%H A323943 S. J. Cyvin, B. N. Cyvin and J. Brunvoll, <a href="https://dx.doi.org/10.1016/0022-2860(95)09039-8">Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons</a>, Journal of molecular structure 376.1-3 (1996): 495-505. See Section 7.3.
%F A323943 T(1,1)=T(1,3)=1, T(1,2)=2; thereafter T(n+1,k) = 3*T(n,k)+T(n,k-1).
%e A323943 Matrix begins:
%e A323943 1, 2, 1,
%e A323943 3, 7, 5, 1,
%e A323943 9, 24, 22, 8, 1,
%e A323943 27, 81, 90, 46, 11, 1,
%e A323943 81, 270, 351, 228, 79, 14, 1,
%e A323943 ...
%p A323943 A323943 := proc(n,k)
%p A323943     option remember;
%p A323943     if n = 1 then
%p A323943         if k>=1 and k<=3 then
%p A323943             op(k,[1,2,1]) ;
%p A323943         else
%p A323943             0;
%p A323943         end if;
%p A323943     else
%p A323943         3*procname(n-1,k)+procname(n-1,k-1) ;
%p A323943     end if;
%p A323943 end proc:
%p A323943 seq(seq(A323943(n,k),k=1..n+2),n=1..12) ; # _R. J. Mathar_, May 08 2019
%t A323943 T[n_, k_] := T[n, k] = If[n == 1, If[k >= 1 && k <= 3, {1, 2, 1}[[k]], 0], 3*T[n - 1, k] + T[n - 1, k - 1]];
%t A323943 Table[Table[T[n, k], {k, 1, n + 2}], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Nov 08 2023, after _R. J. Mathar_ *)
%Y A323943 Cf. A024462 (reversed rows).
%Y A323943 Cf. A000244 (column 1), A038765 (column 2), A081892 (column 3)
%K A323943 nonn,tabf,easy
%O A323943 1,2
%A A323943 _N. J. A. Sloane_, Feb 09 2019