A208331 Triangle of coefficients of polynomials v(n,x) jointly generated with A208330; see the Formula section.

1, 1, 3, 1, 6, 5, 1, 9, 15, 11, 1, 12, 30, 44, 21, 1, 15, 50, 110, 105, 43, 1, 18, 75, 220, 315, 258, 85, 1, 21, 105, 385, 735, 903, 595, 171, 1, 24, 140, 616, 1470, 2408, 2380, 1368, 341, 1, 27, 180, 924, 2646, 5418, 7140, 6156, 3069, 683, 1, 30, 225

Offset

1,3

Comments

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

Links

Table of n, a(n) for n=1..58.

Formula

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

T(n,k) = A001045(n+2)*binomial(n-1,k). - Philippe Deléham, Mar 18 2012

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), T(1,0) = T(2,0) = 1, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 18 2012

Example

First five rows:
1
1...3
1...6...5
1...9...15...11
1...12...30...44...21
First five polynomials u(n,x):
1, 1 + 3x, 1 + 6x + 5x^2, 1 + 9x + 15x^2 + 11x^3, 1+12x + 30x^2 + 44x^3 + 21x^4.
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, ...) begins :
1
1, 0
1, 3, 0
1, 6, 5, 0
1, 9, 15, 11, 0
1, 12, 30, 44, 21, 0. - Philippe Deléham, Mar 18 2012

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 + 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[%]  (* A208330 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208331 *)

Crossrefs

Cf. A208330.

Sequence in context: A308948 A225246 A116666 * A061702 A112351 A143858

Adjacent sequences: A208328 A208329 A208330 * A208332 A208333 A208334

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved