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A209414 Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section. 3
1, 1, 1, 1, 4, 1, 1, 7, 9, 1, 1, 10, 26, 16, 1, 1, 13, 52, 70, 25, 1, 1, 16, 87, 190, 155, 36, 1, 1, 19, 131, 403, 553, 301, 49, 1, 1, 22, 184, 736, 1462, 1372, 532, 64, 1, 1, 25, 246, 1216, 3206, 4446, 3024, 876, 81, 1, 1, 28, 317, 1870, 6190, 11584, 11826, 6084, 1365, 100, 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, -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 01 2012

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1225

FORMULA

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

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

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

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

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

1...7....9....1

1...10...26...16...1

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

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

1

1, 0

1, 1, 0

1, 4, 1, 0

1, 7, 9, 1, 0

1, 10, 26, 16, 1, 0

1, 13, 52, 70, 25, 1, 0 . - Philippe Deléham, Apr 01 2012

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_] := 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[%]    (* A209414 *)

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

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

TableForm[cv]

Flatten[%]    (* A112351 *)

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

CROSSREFS

Cf. A112351, A208510.

Sequence in context: A316123 A146771 A073697 * A193636 A232968 A119673

Adjacent sequences:  A209411 A209412 A209413 * A209415 A209416 A209417

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 09 2012

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)