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Triangular sequence T(n, m) = (p^(n-m)*q^m + p^m*q^(n-m))*A(n+1, m+1), where A(n, m) = (3*n -3*k +1)A(n-1, k-1) + (3*k-2)A(n-1, k), A(n,1)=A(n,n)=1, p=2 and q=3.
1

%I #13 Jun 03 2023 06:27:26

%S 2,5,5,13,96,13,35,1170,1170,35,97,12948,39312,12948,97,275,142170,

%T 986760,986760,142170,275,793,1585368,22077900,47364480,22077900,

%U 1585368,793,2315,18009750,470999340,1846449000,1846449000,470999340,18009750,2315

%N Triangular sequence T(n, m) = (p^(n-m)*q^m + p^m*q^(n-m))*A(n+1, m+1), where A(n, m) = (3*n -3*k +1)A(n-1, k-1) + (3*k-2)A(n-1, k), A(n,1)=A(n,n)=1, p=2 and q=3.

%H G. C. Greubel, <a href="/A154698/b154698.txt">Rows n = 0..20 of triangle, flattened</a>

%H A. Lakhtakia, R. Messier, V. K. Varadan, V. V. Varadan, <a href="http://dx.doi.org/10.1103/PhysRevA.34.2501">Use of combinatorial algebra for diffusion on fractals</a>, Physical Review A, volume 34, Number 3 (1986) p. 2501, (FIG. 3).

%e Triangle begins as:

%e 2;

%e 5, 5;

%e 13, 96, 13;

%e 35, 1170, 1170, 35;

%e 97, 12948, 39312, 12948, 97;

%e 275, 142170, 986760, 986760, 142170, 275;

%e 793, 1585368, 22077900, 47364480, 22077900, 1585368, 793;

%t p=2; q=3;

%t A[n_, 1]:= 1; A[n_, n_]:= 1; A[n_, k_]:= (3*n-3*k+1)*A[n-1, k-1] + (3*k-2)*A[n-1, k];

%t T[n_, m_] := (p^(n-m)*q^m + p^m*q^(n-m)) *A[n+1, m+1];

%t Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten (* modified by _G. C. Greubel_, May 08 2019 *)

%K nonn,tabl

%O 0,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Jan 14 2009

%E Edited by _G. C. Greubel_, May 08 2019