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 A208342 Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section. 5
 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 10, 8, 1, 1, 6, 9, 16, 18, 13, 1, 1, 7, 11, 23, 31, 33, 21, 1, 1, 8, 13, 31, 47, 62, 59, 34, 1, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 1, 11, 19, 61, 113, 213, 321, 414 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Coefficient of x^(n-1): A000045(n) (Fibonacci numbers). n-th row sum: 2^(n-1). Mirror image of triangle in A053538. - Philippe Deléham, Mar 05 2012 Subtriangle of the triangle T(n,k) given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 12 2012 LINKS FORMULA u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = x*u(n-1,x) + x*v(n-1,x), where u(1,x) = 1, v(1,x) = 1. T(n,k) = A208747(n,k)/2^k. - Philippe Deléham, Mar 05 2012 From Philippe Deléham, Mar 12 2012: (Start) As DELTA-triangle T(n,k) with 0<=k<=n: G.f.: (1-y*x+y*x^2-y^2*x^2)/(1-x-y*x+t*x^2-y^2*x^2). T(n,k) = T(n-1,k) + 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) O.g.f.: 1/(1 - z - x*z(1 - z + x*z)) = 1 + (1 + x)*z + (1 + x + 2*x^2)*z^2 + (1 + x + 3*x + 3*x^2)*z^3 + .... - Peter Bala, Dec 31 2015 u(n,x) = Sum_{j=1..floor((n+1)/2)} (-1)^(j-1)*binomial(n-j,j-1)*(x*(1-x))^(j-1)* (1+x)^(n+1-2*j) for n>=1. - Werner Schulte, Mar 07 2017 T(n,k) = Sum_{j=0..floor((k-1)/2)} binomial(k-1-j,j)*binomial(n-k+j,j) for k,n>0 and k<=n (conjectured). - Werner Schulte, Mar 07 2017 EXAMPLE First five rows:   1   1, 1   1, 1, 2   1, 1, 3, 3   1, 1, 4, 5, 5 First five polynomials u(n,x): 1, 1 + x, 1 + x + x^2, 1 + x + 3*x^2 + 3*x^3, 1 + x + 4*x^2 + 5*x^3 + 5*x^4. (1, 0, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins: 1 1, 0 1, 1, 0 1, 1, 2,  0 1, 1, 3,  3,  0 1, 1, 4,  5,  5,  0 1, 1, 5,  7, 10,  8,  0 1, 1, 6,  9, 16, 18, 13,  0 1, 1, 7, 11, 23, 31, 33, 21, 0 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*u[n - 1, x] + 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[%]  (* A208342 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]  (* A208343 *) CROSSREFS Cf. A208343, A208510. Sequence in context: A296313 A183342 A046688 * A157283 A067049 A349976 Adjacent sequences:  A208339 A208340 A208341 * A208343 A208344 A208345 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Feb 25 2012 STATUS approved

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Last modified January 18 07:55 EST 2022. Contains 350454 sequences. (Running on oeis4.)