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A193820 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

%I A193820 #30 Jan 10 2025 10:04:18
%S A193820 1,1,1,1,2,2,1,3,4,4,1,4,7,8,8,1,5,11,15,16,16,1,6,16,26,31,32,32,1,7,
%T A193820 22,42,57,63,64,64,1,8,29,64,99,120,127,128,128,1,9,37,93,163,219,247,
%U A193820 255,256,256,1,10,46,130,256,382,466,502,511,512,512,1,11,56
%N A193820 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.
%C A193820 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%C A193820 Variant of A054143 and A008949. - _R. J. Mathar_, Mar 03 2013
%F A193820 From _Peter Bala_, Jul 16 2013: (Start)
%F A193820 T(n,k) = sum {i = 0..k} binomial(n-1,k-i) for 0 <= k <= n.
%F A193820 O.g.f.: (1 - x*t)^2/( (1 - 2*x*t)*(1 - (1 + x)*t) ) = 1 + (1 + x)*t + (1 + 2*x + 2*x^2)*t^2 + ....
%F A193820 The n-th row polynomial R(n,x) for n >= 1 is given by R(n,x) = 1/(1 - x)*( (x + 1)^(n-1) - 2^(n-1)*x^(n+1) ). Cf. A193823. (End)
%e A193820 First six rows:
%e A193820   1
%e A193820   1....1
%e A193820   1....2....2
%e A193820   1....3....4....4
%e A193820   1....4....7....8....8
%e A193820   1....5....11...15...16...16
%p A193820 A193820 := (n,k) -> `if`(k=0 or n=0,1, A193820(n-1,k-1)+A193820(n-1,k));
%p A193820 seq(print(seq(A193820(n,k),k=0..n+1)),n=0..10); # _Peter Luschny_, Jan 22 2012
%t A193820 z = 10; a = 1; b = 1;
%t A193820 p[n_, x_] := (a*x + b)^n
%t A193820 q[0, x_] := 1
%t A193820 q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
%t A193820 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193820 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193820 g[n_] := CoefficientList[w[n, x], {x}]
%t A193820 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193820 Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193820 *)
%t A193820 TableForm[Table[g[n], {n, -1, z}]]
%t A193820 Flatten[Table[g[n], {n, -1, z}]]  (* A128175 *)
%Y A193820 Cf. A193722, A128175, A193823, A045623 (row sums), A009766.
%Y A193820 Cf. A054143, A008949.
%K A193820 nonn,tabl
%O A193820 0,5
%A A193820 _Clark Kimberling_, Aug 06 2011