A249128 - OEIS

%I #13 Feb 28 2025 07:45:28

%S 1,1,1,1,1,1,2,3,1,1,2,4,5,1,1,6,11,7,8,1,1,6,18,26,10,11,1,1,24,50,

%T 46,58,14,15,1,1,24,96,154,86,102,18,19,1,1,120,274,326,444,156,177,

%U 23,24,1,1,120,600,1044,756,954,246,272,28,29,1,1,720,1764,2556,3708,1692,2016,384,416,34,35,1,1

%N Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

%C The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + floor((n+1)/2)/f(n-1,x), where f(0,x) = 1.

%C (Sum of numbers in row n) = A056953(n) for n >= 0.

%C Column 1 consists of repeated factorials (A000142), as in A081123.

%H Clark Kimberling, <a href="/A249128/b249128.txt">Rows 0..100, flattened</a>

%e f(0,x) = 1/1, so that p(0,x) = 1;

%e f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;

%e f(2,x) = (1 + x + x^2)/(1 + x), so that p(2,x) = 1 + x + x^2.

%e First 6 rows of the triangle of coefficients:

%e   1

%e   1    1

%e   1    1    1

%e   2    3    1    1

%e   2    4    5    1    1

%e   6    11   7    8    1    1

%t z = 15; p[x_, n_] := x + Floor[n/2]/p[x, n - 1]; p[x_, 1] = 1;

%t t = Table[Factor[p[x, n]], {n, 1, z}]

%t u = Numerator[t]

%t TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249128 array *)

%t Flatten[CoefficientList[u, x]] (* A249128 sequence *)

%Y Cf. A056953, A000142, A081123, A249130.

%K nonn,tabl,easy

%O 0,7

%A _Clark Kimberling_, Oct 22 2014