A230698 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.

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, 279, 141, 185, 25, 27, 1, 1, 384, 561, 975, 285, 345, 33, 35, 1, 1, 945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1, 3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1

Offset

0,4

Comments

Triangle A180048 mixed with triangle A180049.

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

Links

Clark Kimberling, Rows 0..100, flattened

Formula

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.

T(n,0) = A006882(n).

T(n+1,1) = A007911(n+3).

Sum_{k=0..n} T(n,k) = A000085(n+1).

Example

Triangle begins (0<=k<=n):
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, 279, 141, 185, 25, 27, 1, 1
384, 561, 975, 285, 345, 33, 35, 1, 1
945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1
3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1
10395, 35685, 26685, 41685, 10290, 12558, 1302, 1422, 63, 65, 1, 1

Mathematica

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 *)
(* 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 *)
Flatten[CoefficientList[u, x]] (* A249057 sequence *)
(* Clark Kimberling, Oct 19 2014 *)

Crossrefs

Cf. A180048, A180049.

Sequence in context: A189435 A279636 A290569 * A090234 A286380 A275866

Adjacent sequences: A230695 A230696 A230697 * A230699 A230700 A230701

Keyword

nonn,tabl

Author

Philippe Deléham, Oct 28 2013

Extensions

Corrected by Clark Kimberling, Oct 21 2014

Status

approved