A208339 Triangle of coefficients of polynomials v(n,x) jointly generated with A208838; see the Formula section.
1, 1, 3, 1, 4, 7, 1, 5, 13, 17, 1, 6, 20, 40, 41, 1, 7, 28, 72, 117, 99, 1, 8, 37, 114, 241, 332, 239, 1, 9, 47, 167, 425, 769, 921, 577, 1, 10, 58, 232, 682, 1492, 2368, 2512, 1393, 1, 11, 70, 310, 1026, 2598, 5008, 7096, 6761, 3363, 1, 12, 83, 402, 1472
Offset: 1
Examples
First five rows: 1 1...3 1...4...7 1...5...13...17 1...6...20...40...41 First five polynomials v(n,x): 1 1 + 3x 1 + 4x + 7x^2 1 + 5x + 13x^2 + 17x^3 1 + 6x + 20x^2 + 40x^3 + 41x^4 Contribution from _Philippe Deléham_, Mar 27 2012: (Start) (1, 0, -2/3, 2/3, 0, 0,...) DELTA (0, 3, -2/3, -1/3, 0, 0,...) begins : 1 1, 0 1, 3, 0 1, 4, 7, 0 1, 5, 13, 17, 0 1, 6, 20, 40, 41, 0. (End)
Crossrefs
Cf. A208338.
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] + 2 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[%] (* A208338 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A208339 *)
Formula
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
Contribution from Philippe Deléham, Mar 27 2012: (Start)
As DELTA-triangle T(n,k) with 0<=k<=n:
G.f.: (1-2*y*x+2*y*x^2-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + 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 nd T(n,k) = 0 if k<0 or if k>n. (End)
G.f.: -(1+x*y)*x*y/(-1+2*x*y-x^2*y+x^2*y^2+x). - R. J. Mathar, Aug 11 2015
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