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A209132 Triangle of coefficients of polynomials v(n,x) jointly generated with A209131; see the Formula section. 3
1, 0, 3, 0, 4, 5, 0, 4, 12, 11, 0, 4, 20, 36, 21, 0, 4, 28, 76, 92, 43, 0, 4, 36, 132, 244, 228, 85, 0, 4, 44, 204, 508, 716, 540, 171, 0, 4, 52, 292, 916, 1732, 1972, 1252, 341, 0, 4, 60, 396, 1500, 3564, 5436, 5196, 2844, 683, 0, 4, 68, 516, 2292, 6564 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

As triangle T(n,k) with 0<=k<=n, it is (0, 4/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

Row sums are powers of 3 (A000244). - Philippe Deléham, Mar 21 2012

LINKS

Table of n, a(n) for n=1..61.

FORMULA

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

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

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

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n.- Philippe Deléham, Mar 21 2012

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

EXAMPLE

First five rows:

1

0...3

0...4...5

0...4...12...11

0...4...20...36...21

First three polynomials v(n,x): 1, 3x, 4x + 5x^2.

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*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[%]    (* A209131 *)

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

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

TableForm[cv]

Flatten[%]    (* A209132 *)

CROSSREFS

Cf. A209131, A208510.

Sequence in context: A016644 A240966 A308109 * A019789 A137335 A011077

Adjacent sequences:  A209129 A209130 A209131 * A209133 A209134 A209135

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified October 20 12:47 EDT 2019. Contains 328257 sequences. (Running on oeis4.)