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A208517
Triangle of coefficients of polynomials v(n,x) jointly generated with A208516; see the Formula section.
3
1, 1, 3, 1, 3, 7, 1, 3, 8, 17, 1, 3, 9, 22, 41, 1, 3, 10, 27, 59, 99, 1, 3, 11, 32, 78, 157, 239, 1, 3, 12, 37, 98, 222, 414, 577, 1, 3, 13, 42, 119, 294, 622, 1084, 1393, 1, 3, 14, 47, 141, 373, 864, 1722, 2821, 3363, 1, 3, 15, 52, 164, 459, 1141, 2500, 4719
OFFSET
1,3
FORMULA
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=x*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
1...3
1...3...7
1...3...8...17
1...3...9...22...41
First five polynomials v(n,x):
1
1 + 3x
1 + 3x + 7x^2
1 + 3x + 8x^2 + 17x^3
1 + 3x + 9x^2 + 22x^3 + 41x^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_] := x*u[n - 1, x] + 2 x*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[%] (* A208516 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208517 *)
CROSSREFS
Cf. A208514.
Sequence in context: A257258 A259325 A094250 * A336203 A209566 A208916
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 28 2012
STATUS
approved