login
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.
4

%I #25 Sep 09 2024 14:16:00

%S 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 22,42,57,63,64,64,1,8,29,64,99,120,127,128,128,1,9,37,93,163,219,247,

%U 255,256,256,1,10,46,130,256,382,466,502,511,512,512,1,11,56

%N 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 See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

%C Variant of A054143 and A008949. - _R. J. Mathar_, Mar 03 2013

%F From _Peter Bala_, Jul 16 2013: (Start)

%F T(n,k) = sum {i = 0..k} binomial(n-1,k-i) for 0 <= k <= n.

%F 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 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 First six rows:

%e 1

%e 1....1

%e 1....2....2

%e 1....3....4....4

%e 1....4....7....8....8

%e 1....5....11...15...16...16

%p A193820 := (n,k) -> `if`(k=0 or n=0,1, A193820(n-1,k-1)+A193820(n-1,k));

%p seq(print(seq(A193820(n,k),k=0..n+1)),n=0..10); # _Peter Luschny_, Jan 22 2012

%t z = 10; a = 1; b = 1;

%t p[n_, x_] := (a*x + b)^n

%t q[0, x_] := 1

%t q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193820 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A128175 *)

%Y Cf. A193722, A128175, A193823, A045623 (row sums).

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Aug 06 2011