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A210198
Triangle of coefficients of polynomials v(n,x) jointly generated with A210197; see the Formula section.
3
1, 3, 1, 7, 4, 15, 12, 1, 31, 32, 6, 63, 80, 24, 1, 127, 192, 80, 8, 255, 448, 240, 40, 1, 511, 1024, 672, 160, 10, 1023, 2304, 1792, 560, 60, 1, 2047, 5120, 4608, 1792, 280, 12, 4095, 11264, 11520, 5376, 1120, 84, 1, 8191, 24576, 28160, 15360, 4032
OFFSET
1,2
COMMENTS
Row sums: A005409
Column 1: -1+2^n
Alternating row sums: 1, 2,3,4,5,6,..., A000027
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=u(n-1,x)+v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
3....1
15...12...1
31...32...6
63...80...24...1
First three polynomials v(n,x): 1, 3 + x , 15 + 12x + x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, 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[%] (* A210197 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210198 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A048739 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A005409 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000217 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)
CROSSREFS
Sequence in context: A269423 A328461 A341494 * A271258 A100584 A185877
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 18 2012
STATUS
approved