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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 (list; table; graph; refs; listen; history; text; internal format)
 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] LINKS 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. Contribution 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 (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 . - Philippe Deléham, Mar 26 2012 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 Cf. A208335, A209415. Sequence in context: A271665 A184049 A125230 * A162430 A305059 A128101 Adjacent sequences:  A208331 A208332 A208333 * A208335 A208336 A208337 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Feb 26 2012 STATUS approved

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Last modified October 16 08:15 EDT 2019. Contains 328051 sequences. (Running on oeis4.)