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A318772 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.

%I A318772 #18 Aug 08 2025 23:29:15
%S A318772 1,1,1,1,1,3,1,6,1,9,1,12,1,15,9,1,18,27,1,21,54,1,24,90,1,27,135,27,
%T A318772 1,30,189,108,1,33,252,270,1,36,324,540,1,39,405,945,81,1,42,495,1512,
%U A318772 405,1,45,594,2268,1215,1,48,702,3240,2835,1,51,819,4455,5670,243,1,54,945,5940,10206,1458
%N A318772 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.
%C A318772 The numbers in rows of the triangle are along a "third layer" skew diagonals pointing top-right in center-justified triangle given in A013610 ((1+3*x)^n) and along a "third layer" skew diagonals pointing top-left in center-justified triangle given in A027465 ((3+x)^n), see links. (Note: First layer of skew diagonals in center-justified triangles of coefficients in expansions of (1+3*x)^n and (3+x)^n are given in A304236 and A304249 respectively.)
%C A318772 The coefficients in the expansion of 1/(1-x-3*x^4) are given by the sequence generated by the row sums.
%C A318772 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.6580980673722..., when n approaches infinity.
%D A318772 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.
%H A318772 G. C. Greubel, <a href="/A318772/b318772.txt">Rows n = 0..120 of the irregular triangle, flattened</a>
%H A318772 Zagros Lalo, <a href="/A318772/a318772.pdf">Third layer skew diagonals in center-justified triangle of coefficients in expansion of (1 + 3x)^n</a>
%H A318772 Zagros Lalo, <a href="/A318772/a318772_1.pdf">Third layer skew diagonals in center-justified triangle of coefficients in expansion of (3 + x)^n</a>
%F A318772 T(n,k) = 3^k * (n - 3*k)!/ ((n - 4*k)! k!), where n >= 0 and 0 <= k <= floor(n/4).
%e A318772 Triangle begins:
%e A318772   1;
%e A318772   1;
%e A318772   1;
%e A318772   1;
%e A318772   1,  3;
%e A318772   1,  6;
%e A318772   1,  9;
%e A318772   1, 12;
%e A318772   1, 15,   9;
%e A318772   1, 18,  27;
%e A318772   1, 21,  54;
%e A318772   1, 24,  90;
%e A318772   1, 27, 135,   27;
%e A318772   1, 30, 189,  108;
%e A318772   1, 33, 252,  270;
%e A318772   1, 36, 324,  540;
%e A318772   1, 39, 405,  945,   81;
%e A318772   1, 42, 495, 1512,  405;
%e A318772   1, 45, 594, 2268, 1215;
%e A318772   ...
%t A318772 T[n_, k_]:= T[n, k]= 3^k(n-3k)!/((n-4k)! k!); Table[T[n, k], {n, 0, 21}, {k, 0, Floor[n/4]} ] // Flatten
%t A318772 T[0, 0] = 1; T[n_, k_] := T[n, k] = If[n<0 || k<0, 0, T[n-1, k] + 3T[n-4, k-1]]; Table[T[n, k], {n, 0, 21}, {k, 0, Floor[n/4]}] // Flatten
%o A318772 (Magma) [3^k*Binomial(n-3*k,k): k in [0..Floor(n/4)], n in [0..24]]; // _G. C. Greubel_, May 12 2021
%o A318772 (Sage) flatten([[3^k*binomial(n-3*k,k) for k in (0..n//4)] for n in (0..24)]) # _G. C. Greubel_, May 12 2021
%Y A318772 Row sums give A318774.
%Y A318772 Cf. A013610, A027465.
%Y A318772 Cf. A304236, A304249.
%Y A318772 Cf. A317496, A317497.
%K A318772 tabf,nonn,easy
%O A318772 0,6
%A A318772 _Zagros Lalo_, Sep 04 2018