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A210791
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Triangle of coefficients of polynomials u(n,x) jointly generated with A210792; see the Formula section.
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3
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1, 1, 1, 1, 2, 2, 1, 3, 7, 3, 1, 4, 17, 14, 5, 1, 5, 36, 42, 30, 8, 1, 6, 72, 104, 111, 58, 13, 1, 7, 141, 233, 329, 251, 111, 21, 1, 8, 275, 494, 862, 848, 553, 206, 34, 1, 9, 538, 1016, 2097, 2479, 2112, 1158, 377, 55, 1, 10, 1058, 2056, 4870, 6608, 6875
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OFFSET
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1,5
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COMMENTS
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Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).
Column 2: 1,2,3,4,5,6,7,8,...
Row sums: A007051.
Alternating row sums: A000129.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012
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LINKS
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Table of n, a(n) for n=1..62.
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FORMULA
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u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x-1)*u(n-1,x) + (x+2)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 29 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 - 2*x - y*x + 2*y*x^2 - y^2*x^2)/(1 - 3*x - y*x + 2*x^2 + 2*y*x^2 - y^2*x^2).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
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EXAMPLE
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First five rows:
1;
1, 1;
1, 2, 2;
1, 3, 7, 3;
1, 4, 17, 14, 5;
First three polynomials u(n,x):
1
1 + x
1 + 2x + 2x^2.
From Philippe Deléham, Mar 29 2012: (Start)
(1, 0, 0, 2, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
1;
1, 0;
1, 1, 0;
1, 2, 2, 0;
1, 3, 7, 3, 0;
1, 4, 17, 14, 5, 0;
1, 5, 36, 42, 30, 8, 0;
1, 6, 72, 104, 111, 58, 13, 0; (End)
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 2; f = 0;
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[%] (* A210791 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210792 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007051 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A001129 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A001333 *)
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CROSSREFS
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Cf. A210792, A208510.
Sequence in context: A158497 A334894 A110564 * A299500 A330141 A007441
Adjacent sequences: A210788 A210789 A210790 * A210792 A210793 A210794
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Mar 26 2012
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STATUS
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approved
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