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A208765 Triangle of coefficients of polynomials u(n,x) jointly generated with A208766; see the Formula section. 3
1, 1, 2, 1, 4, 6, 1, 6, 18, 14, 1, 8, 36, 56, 38, 1, 10, 60, 140, 190, 94, 1, 12, 90, 280, 570, 564, 246, 1, 14, 126, 490, 1330, 1974, 1722, 622, 1, 16, 168, 784, 2660, 5264, 6888, 4976, 1606, 1, 18, 216, 1176, 4788, 11844, 20664, 22392, 14454, 4094, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

FORMULA

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

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

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

From Philippe Deléham, Mar 18 2012: (Start)

As DELTA-triangle with 0 <= k <= n:

G.f.: (1-x-y*x+2*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*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) + 4*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n.

T(n,k) = binomial(n-1,k)*A026597(k). (End)

EXAMPLE

First five rows:

  1;

  1,  2;

  1,  4,  6;

  1,  6, 18, 14;

  1,  8, 36, 56, 38;

First five polynomials u(n,x):

  1

  1 + 2x

  1 + 4x + 6x^2

  1 + 6x + 18x^2 + 14x^3

  1 + 8x + 36x^2 + 56x^3 + 38x^4

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

  1;

  1,  0;

  1,  2,  0;

  1,  4,  6,   0;

  1,  6, 18,  14,   0;

  1,  8, 36,  56,  38,  0;

  1, 10, 60, 140, 190, 94, 0. - Philippe Deléham, Mar 18 2012

MATHEMATICA

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

u[n_, x_] := u[n - 1, x] + 2 x*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[%]     (* A208765 *)

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

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

TableForm[cv]

Flatten[%]     (* A208766 *)

Rest[CoefficientList[CoefficientList[Series[(1-x-y*x+2*y*x^2-4*y^2*x^2)/( 1-2*x-y*x+x^2+y*x^2-4*y^2*x^2), {x, 0, 20}, {y, 0, 20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)

CROSSREFS

Cf. A208766, A208510.

Sequence in context: A033877 A059369 A199530 * A232335 A098473 A121757

Adjacent sequences:  A208762 A208763 A208764 * A208766 A208767 A208768

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 02 2012

STATUS

approved

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Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)