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 A208331 Triangle of coefficients of polynomials v(n,x) jointly generated with A208330; see the Formula section. 3
 1, 1, 3, 1, 6, 5, 1, 9, 15, 11, 1, 12, 30, 44, 21, 1, 15, 50, 110, 105, 43, 1, 18, 75, 220, 315, 258, 85, 1, 21, 105, 385, 735, 903, 595, 171, 1, 24, 140, 616, 1470, 2408, 2380, 1368, 341, 1, 27, 180, 924, 2646, 5418, 7140, 6156, 3069, 683, 1, 30, 225 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012 LINKS 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), where u(1,x)=1, v(1,x)=1. T(n,k) = A001045(n+2)*binomial(n-1,k). - Philippe Deléham, Mar 18 2012 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 18 2012 EXAMPLE First five rows: 1 1...3 1...6...5 1...9...15...11 1...12...30...44...21 First five polynomials u(n,x): 1, 1 + 3x, 1 + 6x + 5x^2, 1 + 9x + 15x^2 + 11x^3, 1+12x + 30x^2 + 44x^3 + 21x^4. (1, 0, 0, 1, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, ...) begins : 1 1, 0 1, 3, 0 1, 6, 5, 0 1, 9, 15, 11, 0 1, 12, 30, 44, 21, 0. - Philippe Deléham, Mar 18 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_] := 2 x*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[%]  (* A208330 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]  (* A208331 *) CROSSREFS Cf. A208330. Sequence in context: A308948 A225246 A116666 * A061702 A112351 A143858 Adjacent sequences:  A208328 A208329 A208330 * A208332 A208333 A208334 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Feb 26 2012 STATUS approved

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Last modified October 20 10:45 EDT 2019. Contains 328257 sequences. (Running on oeis4.)