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 A133084 A007318 * A133080. 6
 1, 2, 1, 3, 2, 1, 4, 3, 4, 1, 5, 4, 10, 4, 1, 6, 5, 20, 10, 6, 1, 7, 6, 35, 20, 21, 6, 1, 8, 7, 56, 35, 56, 21, 8, 1, 9, 8, 84, 56, 126, 56, 36, 8, 1, 10, 9, 120, 84, 252, 126, 120, 36, 10, 1, 11, 10, 165, 120, 462, 252, 330, 120, 55, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums = A003945: (1, 3, 6, 12, 24, 48, 96,...). A133084 is jointly generated with A133567 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+v(n-1,x)+1.  See the Mathematica section. - Clark Kimberling, Feb 28 2012 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened FORMULA Binomial transform of triangle A133080. EXAMPLE First few rows of the triangle are: 1; 2, 1; 3, 2, 1; 4, 3, 4, 1; 5, 4, 10, 4, 1; 6, 5, 20, 10, 6, 1; 7, 6, 35, 20, 21, 6, 1; ... 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_] := x*u[n - 1, x] + 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[%]   (* A133567 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]   (* A133084 *) (* Clark Kimberling, Feb 28 2012 *) T[n_, k_] := If[k == n, 1, (1  - (1 + (-1)^k)/2 )*Binomial[n, k] + ((1 + (-1)^k)/2)*Binomial[n - 1, k - 1]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *) PROG (PARI) for(n=1, 10, for(k=1, n, print1(if(k == n, 1, (1  - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)), ", "))) \\ G. C. Greubel, Oct 21 2017 (MAGMA) /* As triangle */ [[(1-(1+(-1)^k)/2 )*Binomial(n, k)+((1+(-1)^k)/2)*Binomial(n-1, k-1): k in [1..n]]: n in [1.. 11]]; // Vincenzo Librandi, Oct 21 2017 CROSSREFS Cf. A133080, A003945, A133567. Sequence in context: A135227 A104325 A204925 * A118851 A112383 A272464 Adjacent sequences:  A133081 A133082 A133083 * A133085 A133086 A133087 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Sep 16 2007 STATUS approved

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Last modified October 21 11:15 EDT 2019. Contains 328294 sequences. (Running on oeis4.)