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A209136 Triangle of coefficients of polynomials v(n,x) jointly generated with A209135; see the Formula section. 3
1, 1, 3, 1, 8, 5, 1, 15, 23, 11, 1, 24, 66, 66, 21, 1, 35, 150, 240, 165, 43, 1, 48, 295, 678, 747, 404, 85, 1, 63, 525, 1631, 2547, 2157, 947, 171, 1, 80, 868, 3500, 7246, 8560, 5864, 2182, 341, 1, 99, 1356, 6888, 18126, 28018, 26592, 15318, 4929, 683 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...

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

Subtriangle of the triangle given by (1, 0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012

LINKS

Table of n, a(n) for n=1..55.

FORMULA

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

v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),

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

From Philippe Deléham, Apr 11 2012: (Start)

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

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

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

  1,  8,  5;

  1, 15, 23, 11;

  1, 24, 66, 66, 21;

First three polynomials v(n,x):

  1

  1 + 3x

  1 + 8x + 5x^2.

From Philippe Deléham, Apr 11 2012: (Start)

(1, 0, 2/3, 1/3, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, ...) begins:

  1;

  1,   0;

  1,   3,   0;

  1,   8,   5,   0;

  1,  15,  23,  11,   0;

  1,  24,  66,  66,  21,   0;

  1,  35, 150, 240, 165,  43,   0; (End)

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];

v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%]    (* A209135 *)

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

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

TableForm[cv]

Flatten[%]    (* A209136 *)

CROSSREFS

Cf. A209135, A208510.

Sequence in context: A068437 A016477 A172157 * A206800 A258205 A258018

Adjacent sequences:  A209133 A209134 A209135 * A209137 A209138 A209139

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified June 22 04:06 EDT 2021. Contains 345367 sequences. (Running on oeis4.)