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A208334
Triangle of coefficients of polynomials u(n,x) jointly generated with A208335; see the Formula section.
4
1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 10, 11, 6, 1, 1, 15, 25, 21, 7, 1, 1, 21, 50, 57, 30, 9, 1, 1, 28, 91, 133, 99, 45, 10, 1, 1, 36, 154, 280, 275, 168, 58, 12, 1, 1, 45, 246, 546, 675, 523, 250, 78, 13, 1, 1, 55, 375, 1002, 1509, 1433, 885, 370, 95, 15, 1, 1, 66, 550
OFFSET
1,5
COMMENTS
row sums, u(n,1): A000129
row sums, v(n,1): A001333
alternating row sums, u(n,-1): 1,0,-1,-2,-3,-4,-5,-6,...
alternating row sums, v(n,-1): 1,1,1,1,1,1,1,1,1,1,1,...
Subtriangle of the triangle (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 26 2012
Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A209415, i.e., the numbers are the same just read row-wise in the opposite direction. - Christine Bessenrodt, Jul 21 2012
FORMULA
u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 26 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x-y^2*x^2)/(1-2*x-y*x^2+x^2-y^2*x^2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) + 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)
EXAMPLE
First five rows:
1;
1, 1;
1, 3, 1;
1, 6, 4, 1;
1, 10, 11, 6, 1;
First five polynomials u(n,x):
1;
1 + x;
1 + 3x + x^2;
1 + 6x + 4x^2 + x^3;
1 + 10x + 11x^2 + 6x^3 + x^4;
From Philippe Deléham, Mar 26 2012: (Start)
(1, 0, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins:
1;
1, 0;
1, 1, 0;
1, 3, 1, 0;
1, 6, 4, 1, 0;
1, 10, 11, 6, 1, 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_] := (x + 1)*u[n - 1, 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[%] (* A208334 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208335 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)
CROSSREFS
Sequence in context: A271665 A184049 A125230 * A162430 A305059 A355996
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 26 2012
STATUS
approved