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A317026 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 8 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

%I A317026 #23 Aug 08 2025 07:04:58
%S A317026 1,1,1,8,1,16,1,24,64,1,32,192,1,40,384,512,1,48,640,2048,1,56,960,
%T A317026 5120,4096,1,64,1344,10240,20480,1,72,1792,17920,61440,32768,1,80,
%U A317026 2304,28672,143360,196608,1,88,2880,43008,286720,688128,262144,1,96,3520,61440,516096,1835008,1835008
%N A317026 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 8 * T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.
%C A317026 The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013615 ((1+8*x)^n) and along skew diagonals pointing top-left in center-justified triangle given in A038279 ((8+x)^n).
%C A317026 The coefficients in the expansion of 1/(1-x-8*x^2) are given by the sequence generated by the row sums.
%C A317026 The row sums are Generalized Fibonacci numbers (see A015443).
%C A317026 If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 3.3722813232690143..., when n approaches infinity; see A235162 (Decimal expansion of (sqrt(33)+1)/2).
%D A317026 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 70, 98
%H A317026 Zagros Lalo, <a href="/A317026/a317026.pdf">Left-justified triangle</a>
%H A317026 Zagros Lalo, <a href="/A317026/a317026_1.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 8x)^n</a>
%H A317026 Zagros Lalo, <a href="/A317026/a317026_2.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (8 + x)^n</a>
%F A317026 T(n,k) = 8^k*binomial(n-k,k), n >= 0, 0 <= k <= floor(n/2).
%e A317026 Triangle begins:
%e A317026   1;
%e A317026   1;
%e A317026   1, 8;
%e A317026   1, 16;
%e A317026   1, 24, 64;
%e A317026   1, 32, 192;
%e A317026   1, 40, 384, 512;
%e A317026   1, 48, 640, 2048;
%e A317026   1, 56, 960, 5120, 4096;
%e A317026   1, 64, 1344, 10240, 20480;
%e A317026   1, 72, 1792, 17920, 61440, 32768;
%e A317026   1, 80, 2304, 28672, 143360, 196608;
%e A317026   1, 88, 2880, 43008, 286720, 688128, 262144;
%e A317026   1, 96, 3520, 61440, 516096, 1835008, 1835008;
%t A317026 t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 8 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
%t A317026 Table[8^k Binomial[n - k, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten
%o A317026 (GAP) Flat(List([0..13],n->List([0..Int(n/2)],k->8^k*Binomial(n-k,k)))); # _Muniru A Asiru_, Jul 19 2018
%Y A317026 Row sums give A015443.
%Y A317026 Cf. A013615, A038279, A235162.
%K A317026 tabf,nonn,easy
%O A317026 0,4
%A A317026 _Zagros Lalo_, Jul 19 2018