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%I #18 Mar 06 2022 00:09:22
%S 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,
%T 35,56,21,8,1,9,8,84,56,126,56,36,8,1,10,9,120,84,252,126,120,36,10,1,
%U 11,10,165,120,462,252,330,120,55,10,1
%N A007318 * A133080.
%C Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...).
%C 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
%H G. C. Greubel, <a href="/A133084/b133084.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F Binomial transform of triangle A133080.
%e First few rows of the triangle:
%e 1;
%e 2, 1;
%e 3, 2, 1;
%e 4, 3, 4, 1;
%e 5, 4, 10, 4, 1;
%e 6, 5, 20, 10, 6, 1;
%e 7, 6, 35, 20, 21, 6, 1;
%e ...
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A133567 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A133084 *)
%t (* _Clark Kimberling_, Feb 28 2012 *)
%t 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 *)
%o (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
%o (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
%Y Cf. A133080, A003945, A133567.
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_, Sep 16 2007
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