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A209417 Triangle of coefficients of polynomials u(n,x) jointly generated with A209418; see the Formula section. 3
1, 1, 1, 1, 4, 1, 1, 5, 11, 1, 1, 8, 18, 26, 1, 1, 9, 38, 56, 57, 1, 1, 12, 51, 142, 159, 120, 1, 1, 13, 81, 229, 463, 423, 247, 1, 1, 16, 100, 412, 886, 1384, 1072, 502, 1, 1, 17, 140, 584, 1766, 3086, 3896, 2618, 1013, 1, 1, 20, 165, 900, 2850, 6744, 9942, 10494, 6213, 2036, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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

Subtriangle of the triangle given by (1, 0, 2, -3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 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) + 2x*v(n-1,x),

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

Contribution from Philippe Deléham, Apr 01 2012: (Start)

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

G.f.: (1+x-3*y*x-3*y*x^2+2*y^2*x^2)/(1-3*y*x-x^2-y*x^2+2*y^2*x^2).

T(n,k) = 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) -2*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,  4,  1;

  1,  5, 11,  1;

  1,  8, 18, 26,  1;

First three polynomials v(n,x): 1, 1 + x, 1 + 4x + x^2.

Contribution from Philippe Deléham, Apr 01 2012: (Start)

(1, 0, 2, -3, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, ...) begins:

  1;

  1,  0;

  1,  1,  0;

  1,  4,  1,  0;

  1,  5, 11,  1,  0;

  1,  8, 18, 26,  1,  0;

  1,  9, 38, 56, 57,  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] + 2 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[%]    (* A209417 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A209418 *)

CoefficientList[CoefficientList[Series[(1 + x - 3*y*x - y*x^2 + 2*y^2*x^2)/(1 - 3*y*x - x^2 - y*x^2 + 2*y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

CROSSREFS

Cf. A209418, A208510.

Sequence in context: A173077 A131239 A114033 * A267990 A084061 A140262

Adjacent sequences:  A209414 A209415 A209416 * A209418 A209419 A209420

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 09 2012

STATUS

approved

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Last modified October 14 04:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)