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A193823 Triangular array:  the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1. 4
1, 1, 1, 1, 3, 3, 1, 5, 9, 9, 1, 7, 19, 27, 27, 1, 9, 33, 65, 81, 81, 1, 11, 51, 131, 211, 243, 243, 1, 13, 73, 233, 473, 665, 729, 729, 1, 15, 99, 379, 939, 1611, 2059, 2187, 2187, 1, 17, 129, 577, 1697, 3489, 5281, 6305, 6561, 6561, 1, 19, 163, 835, 2851 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

LINKS

Table of n, a(n) for n=0..59.

FORMULA

From Peter Bala, Jul 16 2013: (Start)

T(n,k) = sum {i = 0..k} binomial(n-1,k-i)*2^(k-i) for 0 <= k <= n.

O.g.f.: (1 - 2*x*t)^2/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + x)*t + (1 + 3*x + 3*x^2)*t^2 + .... Cf. A193860.

For n >= 1, the n-th row polynomial R(n,x) = 1/(x-1)*( 3^(n-1)*x^(n+1) - (2*x + 1)^(n-1) ). (End)

EXAMPLE

First six rows:

1

1....1

1....3....3

1....5....9....9

1....7....19...27...27

1....9....33...65...81...81

MATHEMATICA

z = 10; a = 2; b = 1;

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

q[0, x_] := 1

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

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

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

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

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

Flatten[Table[Reverse[g[n]], {n, -1, z}]]   (* A193823 *)

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

Flatten[Table[g[n], {n, -1, z}]]  (* A193824 *)

CROSSREFS

Cf. A193722, A193804. A119258, A193860.

Sequence in context: A164984 A248810 A208610 * A071945 A209583 A144944

Adjacent sequences:  A193820 A193821 A193822 * A193824 A193825 A193826

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 06 2011

STATUS

approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)