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A208328 Triangle of coefficients of polynomials u(n,x) jointly generated with A208329; see the Formula section. 3
1, 1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 9, 11, 1, 1, 9, 13, 25, 21, 1, 1, 11, 17, 43, 53, 43, 1, 1, 13, 21, 65, 97, 125, 85, 1, 1, 15, 25, 91, 153, 255, 273, 171, 1, 1, 17, 29, 121, 221, 441, 597, 609, 341, 1, 1, 19, 33, 155, 301, 691, 1089, 1443, 1325, 683, 1, 1, 21 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

row sums, u(n,1):  A000129

row sums, v(n,1):  A001333

Subtriangle of the triangle T(n,k) given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012

LINKS

Table of n, a(n) for n=1..69.

FORMULA

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

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

Contribution from Philippe Deléham, Mar 07 2012. (Start)

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

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

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

Sum_{k, 0<=k<=n, n>0} T(n,k)*x^k = A000012(n), A000129(n), A083858(n) for x = 0, 1, 2 respectively. (End)

EXAMPLE

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

First five rows:

1

1...1

1...1...3

1...1...5...5

1...1...7...9...11

First five polynomials u(n,x):

1, 1 + x, 1 + x + 3x^2, 1 + x + 5x^2 + 5x^3, 1 + x + 7x^2 + 9x^3 + 11x^4.

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

1

1, 0

1, 1, 0

1, 1, 3, 0

1, 1, 5, 5, 0

1, 1, 7, 9, 11, 0

1, 1, 9, 13, 25, 21, 0

1, 1, 11, 17, 43, 53, 43, 0

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

u[n_, x_] := u[n - 1, x] + x*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[%]  (* A208328 *)

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

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

TableForm[cv]

Flatten[%]  (* A208329 *)

CROSSREFS

Cf. A208329.

Sequence in context: A211315 A096583 A130154 * A134398 A026615 A026681

Adjacent sequences:  A208325 A208326 A208327 * A208329 A208330 A208331

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 26 2012

STATUS

approved

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Last modified October 17 18:58 EDT 2019. Contains 328127 sequences. (Running on oeis4.)