A208335 Triangle of coefficients of polynomials v(n,x) jointly generated with A208834; see the Formula section.
1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 14, 15, 6, 1, 6, 25, 36, 23, 8, 1, 7, 41, 76, 69, 36, 9, 1, 8, 63, 147, 176, 123, 48, 11, 1, 9, 92, 266, 400, 355, 192, 66, 12, 1, 10, 129, 456, 834, 910, 635, 292, 82, 14, 1, 11, 175, 747, 1626, 2131, 1833, 1065, 410, 105, 15, 1
Offset: 1
Examples
First five rows: 1; 2, 1; 3, 3, 1; 4, 7, 5, 1; 5, 14, 15, 6, 1; First five polynomials v(n,x): 1 2 + x 3 + 3x + x^2 4 + 7x + 5x^2 + x^3 5 + 14x + 15x^2 + 6x^3 + x^4 From _Philippe Deléham_, Mar 26 2012: (Start) (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins: 1; 1, 0; 2, 1, 0; 3, 3, 1, 0; 4, 7, 5, 1, 0; 5, 14, 15, 6, 1, 0; (End)
Crossrefs
Cf. A208334.
Programs
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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 *)
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+x^2-y^2*x^2)/(1-2*x+x^2-y*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(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0. (End)
Comments