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A193734 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(2x+1)^n and q(n,x)=(x+2)^n. 4
1, 1, 2, 1, 6, 8, 1, 10, 32, 32, 1, 14, 72, 160, 128, 1, 18, 128, 448, 768, 512, 1, 22, 200, 960, 2560, 3584, 2048, 1, 26, 288, 1760, 6400, 13824, 16384, 8192, 1, 30, 392, 2912, 13440, 39424, 71680, 73728, 32768, 1, 34, 512, 4480, 25088, 93184, 229376 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,...) DELTA (2,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

LINKS

Table of n, a(n) for n=0..51.

FORMULA

T(n,k) = 4*T(n-1,k-1) + T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011

G.f.: (-1+2*x*y)/(-1+4*x*y+x). - R. J. Mathar, Aug 11 2015

EXAMPLE

First six rows:

  1;

  1,  2;

  1,  6,   8;

  1, 10,  32,  32;

  1, 14,  72, 160, 128;

  1, 18, 128, 448, 768, 511;

MATHEMATICA

z = 8; a = 2; b = 1; c = 1; d = 2;

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

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}]] (* A193734 *)

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

Flatten[Table[g[n], {n, -1, z}]]      (* A193735 *)

CROSSREFS

Cf. A084938, A193722, A193735.

Sequence in context: A192232 A243320 A319897 * A318390 A319511 A110608

Adjacent sequences:  A193731 A193732 A193733 * A193735 A193736 A193737

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 04 2011

STATUS

approved

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Last modified February 20 20:34 EST 2020. Contains 332084 sequences. (Running on oeis4.)