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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. 4
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, 22, 42, 57, 63, 64, 64, 1, 8, 29, 64, 99, 120, 127, 128, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256, 256, 1, 10, 46, 130, 256, 382, 466, 502, 511, 512, 512, 1, 11, 56 (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.

Variant of A054143 and A008949. - R. J. Mathar, Mar 03 2013

LINKS

Table of n, a(n) for n=0..68.

FORMULA

From Peter Bala, Jul 16 2013: (Start)

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

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 + ....

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)

EXAMPLE

First six rows:

1

1....1

1....2....2

1....3....4....4

1....4....7....8....8

1....5....11...15...16...16

MAPLE

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

seq(print(seq(A193820(n, k), k=0..n+1)), n=0..10); - Peter Luschny, Jan 22 2012

MATHEMATICA

z = 10; a = 1; 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}]]   (* A193820 *)

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

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

CROSSREFS

Cf. A193722, A128175. A193823.

Sequence in context: A089692 A066201 A303273 * A216368 A123956 A113594

Adjacent sequences:  A193817 A193818 A193819 * A193821 A193822 A193823

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 06 2011

STATUS

approved

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Last modified November 16 22:43 EST 2018. Contains 317275 sequences. (Running on oeis4.)