A209415 Triangle of coefficients of polynomials u(n,x) jointly generated with A209416; see the Formula section.

1, 1, 1, 1, 3, 1, 1, 4, 6, 1, 1, 6, 11, 10, 1, 1, 7, 21, 25, 15, 1, 1, 9, 30, 57, 50, 21, 1, 1, 10, 45, 99, 133, 91, 28, 1, 1, 12, 58, 168, 275, 280, 154, 36, 1, 1, 13, 78, 250, 523, 675, 546, 246, 45, 1, 1, 15, 95, 370, 885, 1433, 1509, 1002, 375, 55, 1, 1, 16, 120, 505, 1435, 2718, 3564, 3135, 1749, 550, 66, 1

Offset

1,5

Comments

For a discussion and guide to related arrays, see A208510.

Subtriangle of the triangle given by (1, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 02 2012

Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A208334, i.e., the numbers are the same just read row-wise in the opposite direction. - Christine Bessenrodt, Jul 21 2012

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

u(n,x) = x*u(n-1,x) + v(n-1,x),

v(n,x) = (x+1)*u(n-1,x) + x*v(n-1,x),

where u(1,x)=1, v(1,x)=1.

From Philippe Deléham, Apr 02 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

G.f.: (1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2).

T(n,k) = 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

Example

First five rows:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  6,  1;
  1,  6, 11, 10,  1;
First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^2.
From Philippe Deléham, Apr 02 2012: (Start)
(1, 0, 1, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:
  1;
  1,   0;
  1,   1,   0;
  1,   3,   1,   0;
  1,   4,   6,   1,   0;
  1,   6,  11,  10,   1,   0;
  1,   7,  21,  25,  15,   1,   0;
  1,   9,  30,  57,  50,  21,   1,   0;
  1,  10,  45,  99, 133,  91,  28,   1,   0; (End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209415 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209416 *)
CoefficientList[CoefficientList[Series[(1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

Crossrefs

Cf. A209416, A208510, A208334.

Sequence in context: A133380 A343168 A105687 * A058879 A208344 A209172

Adjacent sequences: A209412 A209413 A209414 * A209416 A209417 A209418

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 09 2012

Status

approved