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A208761 Triangle of coefficients of polynomials u(n,x) jointly generated with A208762; see the Formula section. 3
1, 1, 2, 1, 6, 4, 1, 12, 18, 8, 1, 20, 52, 50, 16, 1, 30, 120, 186, 126, 32, 1, 42, 240, 534, 576, 306, 64, 1, 56, 434, 1302, 1986, 1654, 718, 128, 1, 72, 728, 2828, 5712, 6632, 4484, 1650, 256, 1, 90, 1152, 5628, 14436, 21912, 20508, 11682, 3726, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Alternating row sums: 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, 1, 0, 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 04 2012

LINKS

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

FORMULA

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

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

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

ecurrence : 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). - Philippe Deléham, Mar 04 2012

G.f.: (-1-x*y+x)*x*y/(-1+x*y+2*x+2*x^2*y^2+x^2*y-x^2). - R. J. Mathar, Aug 12 2015

EXAMPLE

First five rows:

1

1...2

1...6....4

1...12...18...8

1...20...52...50...16

First five polynomials u(n,x):

1

1 + 2x

1 + 6x + 4x^2

1 + 12x + 18x^2 + 8x^3

1 + 20x + 52x^2 + 50x^3 + 16x^4

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

1

1, 0

1, 2, 0

1, 6, 4, 0

1, 12, 18, 8, 0

1, 20, 52, 50, 16, 0

1, 30, 120, 186, 126, 32, 0

... - Philippe Deléham, Mar 04 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_] := (x + 1)*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[%]    (* A208761 *)

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

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

TableForm[cv]

Flatten[%]    (* A208762 *)

CROSSREFS

Cf. A208762, A208510.

Sequence in context: A185045 A208913 A208911 * A123519 A167024 A114687

Adjacent sequences:  A208758 A208759 A208760 * A208762 A208763 A208764

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 03 2012

STATUS

approved

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Last modified October 20 20:24 EDT 2019. Contains 328273 sequences. (Running on oeis4.)