A305838 - OEIS

%I #15 Sep 05 2018 02:27:23

%S 1,1,1,5,1,10,1,15,25,1,20,75,1,25,150,125,1,30,250,500,1,35,375,1250,

%T 625,1,40,525,2500,3125,1,45,700,4375,9375,3125,1,50,900,7000,21875,

%U 18750,1,55,1125,10500,43750,65625,15625,1,60,1375,15000,78750,175000,109375

%N Triangle read by rows: T(0,0)= 1; T(n,k)= T(n-1,k) + 5*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-right in center-justified triangle given in A013612 ((1+5*x)^n). The coefficients in the expansion of 1/(1-x-5x^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 2.7913327..., when n approaches infinity. The row sums are A015440 (generalized Fibonacci numbers).

%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 380, 381.

%H Shara Lalo, <a href="/A305838/a305838.pdf">Skew diagonals in triangle A013612</a>

%H Shara Lalo, <a href="/A305838/a305838_1.pdf">Right-justified triangle</a>

%F G.f.: 1/(1 - t*x - 5*t^2).

%e Triangle begins:

%e   1;

%e   1;

%e   1,  5;

%e   1, 10;

%e   1, 15,   25;

%e   1, 20,   75;

%e   1, 25,  150,   125;

%e   1, 30,  250,   500;

%e   1, 35,  375,  1250,    625;

%e   1, 40,  525,  2500,   3125;

%e   1, 45,  700,  4375,   9375,    3125;

%e   1, 50,  900,  7000,  21875,   18750;

%e   1, 55, 1125, 10500,  43750,   65625,   15625;

%e   1, 60, 1375, 15000,  78750,  175000,  109375;

%e   1, 65, 1650, 20625, 131250,  393750,  437500,   78125;

%e   1, 70, 1950, 27500, 206250,  787500, 1312500,  625000;

%e   1, 75, 2275, 35750, 309375, 1443750, 3281250, 2812500,  390625;

%e   1, 80, 2625, 45500, 446875, 2475000, 7218750, 9375000, 3515625;

%t t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0,  t[n - 1, k] + 5 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten

%Y Row sums give A015440.

%Y Cf. A000012 (column 0), A008587 (column 1), A123296 (column 2), A141480 (column 3).

%Y Cf. A013612.

%K tabf,nonn,easy

%O 0,4

%A _Shara Lalo_, Jun 11 2018