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A209413 Triangle of coefficients of polynomials v(n,x) jointly generated with A209172; see the Formula section. 3
1, 1, 2, 1, 3, 4, 1, 5, 7, 8, 1, 6, 17, 15, 16, 1, 8, 23, 49, 31, 32, 1, 9, 39, 72, 129, 63, 64, 1, 11, 48, 150, 201, 321, 127, 128, 1, 12, 70, 198, 501, 522, 769, 255, 256, 1, 14, 82, 338, 699, 1524, 1291, 1793, 511, 512, 1, 15, 110, 420, 1375, 2223, 4339, 3084, 4097, 1023, 1024 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

For n>1, n-th alternating row sum = ((-1)^(n-1)*F(2n-3), where F=A000045 (Fibonacci numbers).

Coefficient of x^(n-1) in u(n,x): 2^(n-1).

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

Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 11 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) = u(n-1,x) + 2x*v(n-1,x),

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

Contribution from Philippe Deléham, Mar 11 2012: (Start)

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

T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(2,0) = 1, T(2,1) = 2, T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n.

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

EXAMPLE

First five rows:

  1;

  1,  2;

  1,  3,  4;

  1,  5,  7,  8;

  1,  6, 17, 15, 16;

First three polynomials v(n,x): 1, 1 + 2x, 1 + 3x + 4x^2.

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

  1;

  1,   0;

  1,   2,   0;

  1,   3,   4,   0;

  1,   5,   7,   8,   0;

  1,   6,  17,  15,  16,   0;

  1,   8,  23,  49,  31,  32,   0;

  1,   9,  39,  72, 129,  63,  64,   0;

  1,  11,  48, 150, 201, 321, 127, 128,   0;

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_] := 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[%]    (* A209172 *)

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

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

TableForm[cv]

Flatten[%]    (* A209413 *)

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

CROSSREFS

Cf. A209172, A208510.

Sequence in context: A224823 A078753 A119443 * A126198 A055888 A094442

Adjacent sequences:  A209410 A209411 A209412 * A209414 A209415 A209416

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 08 2012

STATUS

approved

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Last modified May 19 17:48 EDT 2019. Contains 323395 sequences. (Running on oeis4.)