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A193738 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=q(n,x)=x^n+x^(n-1)+...+x+1.

%I A193738 #8 May 11 2013 18:13:24
%S A193738 1,1,1,1,2,2,1,2,3,3,1,2,3,4,4,1,2,3,4,5,5,1,2,3,4,5,6,6,1,2,3,4,5,6,
%T A193738 7,7,1,2,3,4,5,6,7,8,8,1,2,3,4,5,6,7,8,9,9,1,2,3,4,5,6,7,8,9,10,10,1,
%U A193738 2,3,4,5,6,7,8,9,10,11,11,1,2,3,4,5,6,7,8,9,10,11,12,12,1,2,3,4
%N A193738 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=q(n,x)=x^n+x^(n-1)+...+x+1.
%C A193738 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%H A193738 Reinhard Zumkeller, <a href="/A193738/b193738.txt">Rows n = 0..100 of triangle, flattened</a>
%e A193738 First six rows:
%e A193738 1
%e A193738 1....1
%e A193738 1....2....2
%e A193738 1....2....3....3
%e A193738 1....2....3....4...4
%e A193738 1....2....3....4...5...5
%t A193738 z = 12;
%t A193738 p[0, x_] := 1
%t A193738 p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0
%t A193738 q[n_, x_] := p[n, x]
%t A193738 t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
%t A193738 t[n_, n_] := p[n, x] /. x -> 0;
%t A193738 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193738 g[n_] := CoefficientList[w[n, x], {x}]
%t A193738 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193738 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193738 *)
%t A193738 TableForm[Table[g[n], {n, -1, z}]]
%t A193738 Flatten[Table[g[n], {n, -1, z}]]   (* A193739 *)
%o A193738 (Haskell)
%o A193738 a193738 n k = a193738_tabl !! n !! k
%o A193738 a193738_row n = a193738_tabl !! n
%o A193738 a193738_tabl = map reverse a193739_tabl
%o A193738 -- _Reinhard Zumkeller_, May 11 2013
%Y A193738 Cf. A193722, A193739.
%K A193738 nonn,tabl
%O A193738 0,5
%A A193738 _Clark Kimberling_, Aug 04 2011