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%I #13 Sep 05 2018 02:27:19
%S 1,5,25,1,125,10,625,75,1,3125,500,15,15625,3125,150,1,78125,18750,
%T 1250,20,390625,109375,9375,250,1,1953125,625000,65625,2500,25,
%U 9765625,3515625,437500,21875,375,1,48828125,19531250,2812500,175000,4375,30,244140625,107421875,17578125,1312500,43750,525,1
%N Triangle read by rows: T(0,0) = 1; T(n,k) = 5*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.
%C The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013612 ((1+5*x)^n).
%C The coefficients in the expansion of 1/(1-5x-x^2) are given by the sequence generated by the row sums.
%C If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 5.1925824035..., a metallic mean (see A098318), when n approaches infinity.
%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 92, 380, 382.
%H Shara Lalo, <a href="/A305837/a305837.pdf">Left-justified triangle </a>
%H Shara Lalo, <a href="/A305837/a305837_1.pdf">Skew diagonals in triangle A013612</a>
%F G.f.: 1/(1 - 5*t*x - t^2).
%e Triangle begins:
%e 1;
%e 5;
%e 25, 1;
%e 125, 10;
%e 625, 75, 1;
%e 3125, 500, 15;
%e 15625, 3125, 150, 1;
%e 78125, 18750, 1250, 20;
%e 390625, 109375, 9375, 250, 1;
%e 1953125, 625000, 65625, 2500, 25;
%e 9765625, 3515625, 437500, 21875, 375, 1;
%e 48828125, 19531250, 2812500, 175000, 4375, 30;
%e 244140625, 107421875, 17578125, 1312500, 43750, 525, 1;
%e 1220703125, 585937500, 107421875, 9375000, 393750, 7000, 35;
%e 6103515625, 3173828125, 644531250, 64453125, 3281250, 78750, 700, 1;
%e 30517578125, 17089843750, 3808593750, 429687500, 25781250, 787500, 10500, 40;
%t t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 5 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten
%Y Row sums give A052918.
%Y Cf. A000351 (column 0), A053464 (column 1), A081135 (column 2), A081143 (column 3), A036071 (column 4).
%Y Cf. A013612.
%Y Cf. A098318.
%K tabf,nonn,easy
%O 0,2
%A _Shara Lalo_, Jun 11 2018
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