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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
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
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),
where u(1,x)=1, v(1,x)=1.
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..n, n>0} T(n,k)*x^k = A000012(n), A000129(n), A083858(n) for x = 0, 1, 2 respectively. (End)
EXAMPLE
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.
From Philippe Deléham, Mar 07 2012: (Start)
(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; (End)
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 26 2012
STATUS
approved