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A317497 Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.

%I A317497 #13 Sep 08 2022 08:46:22
%S A317497 1,3,9,27,1,81,6,243,27,729,108,1,2187,405,9,6561,1458,54,19683,5103,
%T A317497 270,1,59049,17496,1215,12,177147,59049,5103,90,531441,196830,20412,
%U A317497 540,1,1594323,649539,78732,2835,15,4782969,2125764,295245,13608,135,14348907,6908733,1082565,61236,945,1
%N A317497 Triangle T(n,k) = 3*T(n-1,k) +  T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.
%C A317497 The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n) and  along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in triangles of coefficients in expansions of (1+3*x)^n and (3+x)^n are given in A304236 and A304249 respectively.)
%C A317497 The coefficients in the expansion of 1/(1-3x-x^3) are given by the sequence generated by the row sums.
%C A317497 The row sums give A052541.
%C A317497 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.1038034027355..., when n approaches infinity.
%D A317497 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 364-366
%H A317497 G. C. Greubel, <a href="/A317497/b317497.txt">Rows n = 0..120 of the irregular triangle, flattened</a>
%H A317497 Zagros Lalo, <a href="/A317497/a317497.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3x)^n</a>
%H A317497 Zagros Lalo, <a href="/A317497/a317497_1.pdf">Second layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n</a>
%F A317497 T(n,k) = 3^(n-3*k) * (n-2*k)!/(k! * (n-3*k)!) where n is a nonnegative integer and k = 0..floor(n/3).
%e A317497 Triangle begins:
%e A317497          1;
%e A317497          3;
%e A317497          9;
%e A317497         27,        1;
%e A317497         81,        6;
%e A317497        243,       27;
%e A317497        729,      108,       1;
%e A317497       2187,      405,       9;
%e A317497       6561,     1458,      54;
%e A317497      19683,     5103,     270,      1;
%e A317497      59049,    17496,    1215,     12;
%e A317497     177147,    59049,    5103,     90;
%e A317497     531441,   196830,   20412,    540,    1;
%e A317497    1594323,   649539,   78732,   2835,   15;
%e A317497    4782969,  2125764,  295245,  13608,  135;
%e A317497   14348907,  6908733, 1082565,  61236,  945,  1;
%e A317497   43046721, 22320522, 3897234, 262440, 5670, 18;
%t A317497 T[n_, k_]:= T[n, k] = 3^(n-3k)(n-2k)!/((n-3k)! k!); Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/3]} ]//Flatten
%t A317497 T[0, 0] = 1; T[n_, k_]:= T[n, k] = If[n<0 || k<0, 0, 3 T[n-1, k] + T[n-3, k-1]]; Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/3]}]//Flatten
%o A317497 (GAP) Flat(List([0..16],n->List([0..Int(n/3)],k->3^(n-3*k)/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # _Muniru A Asiru_, Aug 01 2018
%o A317497 (Magma) [3^(n-3*k)*Binomial(n-2*k,k): k in [0..Floor(n/3)], n in [0..24]]; // _G. C. Greubel_, May 12 2021
%o A317497 (Sage) flatten([[3^(n-3*k)*binomial(n-2*k,k) for k in (0..n//3)] for n in (0..24)]) # _G. C. Greubel_, May 12 2021
%Y A317497 Row sums give A052541.
%Y A317497 Cf. A013610, A027465, A317496, A304236, A304249.
%Y A317497 Cf. A000244 (column 0), A027471 (column 1), A027472 (column 2), A036216 (column 3), A036217 (column 4).
%Y A317497 Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), A304249 (q=2), this sequence (q=3), A318773 (q=4).
%K A317497 tabf,nonn,easy
%O A317497 0,2
%A A317497 _Zagros Lalo_, Jul 31 2018