%I #28 Dec 30 2014 09:55:40
%S 1,1,1,2,1,1,3,5,1,1,8,7,9,1,1,15,33,12,14,1,1,48,57,87,18,20,1,1,105,
%T 279,141,185,25,27,1,1,384,561,975,285,345,33,35,1,1,945,2895,1830,
%U 2640,510,588,42,44,1,1,3840,6555,12645,4680,6090,840,938,52,54,1,1
%N Triangle read by rows: T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.
%C Triangle A180048 mixed with triangle A180049.
%C Let p(n,x) be the polynomial whose coefficients are given by row n; e.g., p(2,x) = 2 + x + x^2; then p(n,x) is the numerator of the rational function given by f(n,x) = x + (n - 1)/f(n-1,x), where f(x,0) = 1. (Sum of numbers in row n) = A000885(n) for n >= 1. (Column 1) = A006882 (n-th term = n!! for n >= 0) - _Clark Kimberling_, Oct 19 2014
%H Clark Kimberling, <a href="/A230698/b230698.txt">Rows 0..100, flattened</a>
%F T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.
%F T(n,0) = A006882(n).
%F T(n+1,1) = A007911(n+3).
%F Sum_{k=0..n} T(n,k) = A000085(n+1).
%e Triangle begins (0<=k<=n):
%e 1
%e 1, 1
%e 2, 1, 1
%e 3, 5, 1, 1
%e 8, 7, 9, 1, 1
%e 15, 33, 12, 14, 1, 1
%e 48, 57, 87, 18, 20, 1, 1
%e 105, 279, 141, 185, 25, 27, 1, 1
%e 384, 561, 975, 285, 345, 33, 35, 1, 1
%e 945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1
%e 3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1
%e 10395, 35685, 26685, 41685, 10290, 12558, 1302, 1422, 63, 65, 1, 1
%t t[0, 0] = 1; t[1, 0] = 1; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k > n || k < 0, 0, t[n - 1, k - 1] + n*t[n - 2, k]]; Table[t[n, k], {n, 0, 10}, {k, 0, n}](* _Clark Kimberling_, Oct 19 2014 *)
%t (* Next, the polynomials *); z = 20; f[x_, n_] := x + n/f[x, n - 1]; f[x_, 0] = 1; t = Table[Factor[f[x, n]], {n, 0, z}]; u = Numerator[t]; TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]] (* A249057 array *)
%t Flatten[CoefficientList[u, x]] (* A249057 sequence *)
%t (* _Clark Kimberling_, Oct 19 2014 *)
%Y Cf. A180048, A180049.
%K nonn,tabl
%O 0,4
%A _Philippe Deléham_, Oct 28 2013
%E Corrected by _Clark Kimberling_, Oct 21 2014