A208340 Triangle of coefficients of polynomials v(n,x) jointly generated with A202390; see the Formula section.
1, 2, 2, 3, 6, 3, 4, 13, 14, 5, 5, 24, 41, 30, 8, 6, 40, 96, 109, 60, 13, 7, 62, 196, 308, 262, 116, 21, 8, 91, 364, 743, 868, 590, 218, 34, 9, 128, 630, 1604, 2413, 2240, 1267, 402, 55, 10, 174, 1032, 3186, 5926, 7046, 5424, 2627, 730, 89, 11, 230, 1617
Offset: 1
Examples
First five rows: 1; 2, 2; 3, 6, 3; 4, 13, 14, 5; 5, 24, 41, 30, 8; The first five polynomials v(n,x): 1 2 + 2x 3 + 6x + 3x^2 4 + 13x + 14x^2 + 5x^3 5 + 24x + 41x^2 + 30x^3 + 8x^4
Crossrefs
Cf. A202390.
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] + (x + 1)*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[%] (* A202390 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A208340 *) 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) + (x+1)*v(n-1,x), where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 28 2012: (Start)
As triangle T(n,k) with 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-2) with T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+y*x)/(1-2*x-y*x+x^2-y^2*x^2).
Comments