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A193862 Mirror of the triangle A115068.

%I A193862 #12 Apr 10 2020 12:38:07
%S A193862 1,2,2,3,6,4,4,12,16,8,5,20,40,40,16,6,30,80,120,96,32,7,42,140,280,
%T A193862 336,224,64,8,56,224,560,896,896,512,128,9,72,336,1008,2016,2688,2304,
%U A193862 1152,256,10,90,480,1680,4032,6720,7680,5760,2560,512,11,110,660
%N A193862 Mirror of the triangle A115068.
%C A193862 A193862 is obtained by reversing the rows of the triangle A115068.
%C A193862 Riordan array (1/(1-x)^2, 2*x/(1-x)). - _Philippe Deléham_, Jan 29 2014
%C A193862 Let P(n, x) := Sum_{k=1..n} T(n, k)*x^k. Then P(n, P(m, x)) = P(n*m, x) for all n and m in Z. - _Michael Somos_, Apr 10 2020
%F A193862 Write w(n,k) for the triangle at A115068.  The triangle at A193862 is then given by w(n,n-k).
%F A193862 T(n, k) = binomial(n, k)/2 * 2^k. - _Michael Somos_, Apr 10 2020
%e A193862 First six rows:
%e A193862 1
%e A193862 2...2
%e A193862 3...6....4
%e A193862 4...12...16...8
%e A193862 5...20...40...40....16
%e A193862 6...30...80...120...96...32
%e A193862 Production matrix begins
%e A193862 2......2
%e A193862 -1/2...1...2
%e A193862 1/4....0...1...2
%e A193862 -1/8...0...0...1...2
%e A193862 1/16...0...0...0...1...2
%e A193862 -1/32..0...0...0...0...1...2
%e A193862 1/64...0...0...0...0...0...1...2
%e A193862 -1/128.0...0...0...0...0...0...1...2
%e A193862 1/256..0...0...0...0...0...0...0...1...2
%e A193862 - _Philippe Deléham_, Jan 29 2014
%t A193862 z = 11;
%t A193862 p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + 1;
%t A193862 q[n_, x_] := (2 x + 1)^n;
%t A193862 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193862 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193862 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193862 h[n_] := CoefficientList[d[n, x], {x}]
%t A193862 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193862 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A115068 *)
%t A193862 TableForm[Table[h[n], {n, 0, z}]]
%t A193862 Flatten[Table[h[n], {n, -1, z}]]   (* A193862 *)
%t A193862 T[ n_, k_] := Binomial[n, k]/2 2^k; (* _Michael Somos_, Apr 10 2020 *)
%o A193862 (PARI) {T(n, k) = binomial(n, k)/2 * 2^k}; /* _Michael Somos_, Apr 10 2020 */
%Y A193862 Cf. A115068.
%K A193862 nonn,tabl
%O A193862 0,2
%A A193862 _Clark Kimberling_, Aug 07 2011