A208524 Triangle of coefficients of polynomials u(n,x) jointly generated with A208525; see the Formula section.
1, 1, 1, 1, 3, 3, 1, 6, 10, 5, 1, 10, 22, 23, 11, 1, 15, 40, 65, 60, 21, 1, 21, 65, 145, 195, 137, 43, 1, 28, 98, 280, 490, 518, 322, 85, 1, 36, 140, 490, 1050, 1484, 1372, 723, 171, 1, 45, 192, 798, 2016, 3570, 4368, 3447, 1624, 341, 1, 55, 255, 1230, 3570
Offset: 1
Examples
First five rows: 1 1...1 1...3....3 1...6....10...5 1...10...22...23...11 First five polynomials u(n,x): 1 1 + x 1 + 3x + 3x^2 1 + 6x + 10x^2 + 5x^3 1 + 10x + 22x^2 + 23x^3 + 11x^4
Crossrefs
Cf. A208525.
Programs
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Mathematica
u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1; 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[%] (* A208524 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A208525 *) Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*) Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*) Table[u[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*) Table[v[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)
Formula
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Comments