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A193895 Triangular array:  the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{(k+1)*x^(n-k) : 0<=k<=n} and q(n,x)=sum{(k+1)*x^k : 0<=k<=n}. 2

%I

%S 1,2,1,6,6,3,12,15,12,6,20,28,27,20,10,30,45,48,42,30,15,42,66,75,72,

%T 60,42,21,56,91,108,110,100,81,56,28,72,120,147,156,150,132,105,72,36,

%U 90,153,192,210,210,195,168,132,90,45,110,190,243,272,280,270

%N Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{(k+1)*x^(n-k) : 0<=k<=n} and q(n,x)=sum{(k+1)*x^k : 0<=k<=n}.

%C See A193722 for the definition of fusion of P by Q (two sequences of polynomials or triangular arrays).

%C ...

%C First six rows of P, the coefficients of (p(n,x)):

%C 1

%C 1...2

%C 1...2...3

%C 1...2...3...4

%C 1...2...3...4...5

%C ...

%C First six rows of Q, the coefficients of (q(n,x)):

%C 1

%C 2...1

%C 3...2...1

%C 4...3...2...1

%C 5...4..3...2..1

%e First six rows of A193895:

%e 1

%e 2....1

%e 6....6....3

%e 12...15...12...6

%e 20...28...27...20...10

%e 30...45...48...42...30...15

%t z = 9;

%t p[n_, x_] := x*p[n - 1, x] + n + 1 (* #6 *) ; p[0, x_] := 1;

%t q[n_, x_] := (n + 1)*x^n + q[n - 1, x] (* #7 *); q[0, x_] := 1;

%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}]] (* A193895 *)

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

%t Flatten[Table[g[n], {n, -1, z}]] (* A193896 *)

%Y A193722, A193896.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 08 2011

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Last modified October 22 04:53 EDT 2021. Contains 348160 sequences. (Running on oeis4.)