A208524 Triangle of coefficients of polynomials u(n,x) jointly generated with A208525; see the Formula section.

1, 1, 1, 1, 3, 3, 1, 6, 10, 5, 1, 10, 22, 23, 11, 1, 15, 40, 65, 60, 21, 1, 21, 65, 145, 195, 137, 43, 1, 28, 98, 280, 490, 518, 322, 85, 1, 36, 140, 490, 1050, 1484, 1372, 723, 171, 1, 45, 192, 798, 2016, 3570, 4368, 3447, 1624, 341, 1, 55, 255, 1230, 3570

Offset

1,5

Comments

Alternating row sums: 1,0,1,0,1,0,1,0,...

Links

Table of n, a(n) for n=1..60.

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)+1,

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

Example

First five rows:
1
1...1
1...3....3
1...6....10...5
1...10...22...23...11
First five polynomials u(n,x):
1
1 + x
1 + 3x + 3x^2
1 + 6x + 10x^2 + 5x^3
1 + 10x + 22x^2 + 23x^3 + 11x^4

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
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] + 1;
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[%]  (* A208524 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208525 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*)
Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*)
Table[u[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)
Table[v[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)

Crossrefs

Cf. A208525.

Sequence in context: A110640 A143389 A219218 * A094040 A039798 A193560

Adjacent sequences: A208521 A208522 A208523 * A208525 A208526 A208527

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 29 2012

Status

approved