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A208910 Triangle of coefficients of polynomials v(n,x) jointly generated with A208755; see the Formula section. 2
1, 1, 3, 1, 3, 8, 1, 3, 10, 22, 1, 3, 12, 32, 60, 1, 3, 14, 42, 100, 164, 1, 3, 16, 52, 144, 308, 448, 1, 3, 18, 62, 192, 480, 936, 1224, 1, 3, 20, 72, 244, 680, 1568, 2816, 3344, 1, 3, 22, 82, 300, 908, 2352, 5040, 8400, 9136, 1, 3, 24, 92, 360, 1164, 3296 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012
LINKS
FORMULA
u(n,x) = u(n-1,x) + 2x*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.
From Philippe Deléham, Apr 01 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 - 2*y*x + 3*y*x^2 - 2*y^2*x^2)/(1 - x - 2*y*x + 2*y*x^2 - 2*y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
T(n,k) = 2*A208757(n,k) - A208332(n,k). - Philippe Deléham, Apr 15 2012
EXAMPLE
First five rows:
1;
1, 3;
1, 3, 8;
1, 3, 10, 22;
1, 3, 12, 32, 60;
First five polynomials v(n,x):
1
1 + 3x
1 + 3x + 8x^2
1 + 3x + 10x^2 + 22x^3
1 + 3x + 12x^2 + 32x^3 + 60x^4
From Philippe Deléham, Apr 01 2012: (Start)
(1, 0, -1, 1, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, ...) begins:
1;
1, 0;
1, 3, 0;
1, 3, 8, 0;
1, 3, 10, 22, 0;
1, 3, 12, 32, 60, 0;
1, 3, 14, 42, 100, 164, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 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[%] (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208910 *)
CROSSREFS
Sequence in context: A132476 A328807 A103279 * A209760 A046544 A011088
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 03 2012
STATUS
approved

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Last modified March 28 08:19 EDT 2024. Contains 371236 sequences. (Running on oeis4.)