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 A208338 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section. 3
 1, 1, 1, 1, 2, 3, 1, 3, 7, 7, 1, 4, 12, 20, 17, 1, 5, 18, 40, 57, 41, 1, 6, 25, 68, 129, 158, 99, 1, 7, 33, 105, 243, 399, 431, 239, 1, 8, 42, 152, 410, 824, 1200, 1160, 577, 1, 9, 52, 210, 642, 1506, 2692, 3528, 3089, 1393, 1, 10, 63, 280, 952, 2532, 5290 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012 LINKS 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. From Philippe Deléham, Apr 09 2012: (Start) As DELTA-triangle T(n,k) with 0 <= k <= n: G.f.: (1-2*y*x-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) = 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,  2,  3;   1,  3,  7,  7;   1,  4, 12, 20, 17; First five polynomials u(n,x):   1   1 +  x   1 + 2x +  3x^2   1 + 3x +  7x^2 +  7x^3   1 + 4x + 12x^2 + 20x^3 + 17x^4 From Philippe Deléham, Apr 09 2012: (Start) (1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, ...) begins:   1;   1,  0;   1,  1,  0;   1,  2,  3,  0;   1,  3,  7,  7,  0;   1,  4, 12, 20, 17,  0;   1,  5, 18, 40, 57, 41,  0; (End) 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 *) CROSSREFS Cf. A208339. Sequence in context: A063967 A059397 A209567 * A236918 A152821 A071943 Adjacent sequences:  A208335 A208336 A208337 * A208339 A208340 A208341 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Feb 27 2012 STATUS approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)