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A104709 Triangle read by rows: T(n,k) = Sum_{j=0..n} 2^(n-j)*binomial(j,k); Riordan array (1/((1-x)(1-2x)), x/(1-x)). 5
1, 3, 1, 7, 4, 1, 15, 11, 5, 1, 31, 26, 16, 6, 1, 63, 57, 42, 22, 7, 1, 127, 120, 99, 64, 29, 8, 1, 255, 247, 219, 163, 93, 37, 9, 1, 511, 502, 466, 382, 256, 130, 46, 10, 1, 1023, 1013, 968, 848, 638, 386, 176, 56, 11, 1, 2047, 2036, 1981, 1816, 1486, 1024, 562, 232, 67 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A104709 is the mirror of the fission, A054143, of the polynomial sequence ((x+1^n) by the polynomial sequence (q(n,x)) given by q(n,x) = x^n + x^(n-1) + ... + x + 1.  See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011

The elements of the matrix inverse appear to be T^(-1)(n,k) = (-1)^(n+k)*A110813(n,k) assuming the same offset in both triangles. - R. J. Mathar, Mar 15 2013

From Paul Curtz, Jun 12 2019: (Start)

Numerators of the triangle [Curtz, page 15, triangle (E)]:

     1/2;

     3/4,  1/4;

     7/8,  4/8,    1/8;

   15/16, 11/16,  5/16,  1/16;

   31/32, 26/31, 16/32,  6/32, 1/32;

   63/64, 57/64, 42/64, 22/64, 7/64, 1/64;

   ... .

Denominators - Numerators: Triangle A054143.

  1;

  1, 3;

  1, 4, 7;

  1, 5, 11, 15;

  ...

(E) is a transform which accelerates the convergence of series.

For log(2) = 1 - 1/2 + 1/3 - 1/4 ... = 0.6931..., we have

  1*(1/2) = 1/2,

  1*(3/4) - (1/2)*(1/4) = 5/8,

  1*(7/8) - (1/2)*(4/8) + (1/3)*(1/8) = 2/3,

  1*(15/16) - (1/2)*(11/16) + (1/3)*(5/16) - (1/4)*1/16 = 131/192,

  ... .

This is A068566/A068565.

(End)

LINKS

Table of n, a(n) for n=1..64.

Paul Curtz, Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.

FORMULA

Begin with A055248 as a triangle, delete leftmost column.

Factors as (1/(1-2x), x)*(1/(1-x), x/(1-x)) - the sequence array for 2^n times Pascal's triangle. - Paul Barry, Aug 05 2005

T(n,k) = Sum_{j=0..n-k} C(n-j, k)*2^j. - Paul Barry, Jan 12 2006

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1), T(0,0)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 30 2013

Working with an offset of 0, we have exp(x) * (e.g.f. for row n) = (e.g.f. for diagonal n). For example, for n = 3 we have exp(x)*(15 + 11*x + 5*x^2/2! + x^3/3!) = 15 + 26*x + 42*x^2/2! + 64*x^3/3! + 93*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 21 2014

EXAMPLE

The first few rows are:

   1;

   3,  1;

   7,  4,  1;

  15, 11,  5,  1;

  31, 26, 16,  6,  1;

  63, 57, 42, 22,  7,  1;

  ...

MAPLE

A104709_row := proc(n) add(add(binomial(n, n-i)*x^(n-k-1), i=0..k), k=0..n-1);

coeffs(sort(%)) end; seq(print(A104709_row(n)), n=1..6); # Peter Luschny, Sep 29 2011

MATHEMATICA

z = 10;

p[n_, x_] := (x + 1)^n;

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

p1[n_, k_] := Coefficient[p[n, x], x^k];

p1[n_, 0] := p[n, x] /. x -> 0;

d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]

h[n_] := CoefficientList[d[n, x], {x}]

TableForm[Table[Reverse[h[n]], {n, 0, z}]]

Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A054143 *)

TableForm[Table[h[n], {n, 0, z}]]

Flatten[Table[h[n], {n, -1, z}]] (* A104709 *)

(* Clark Kimberling, Aug 07 2011 *)

CROSSREFS

Cf. A001787 (row sums), A055248, A007318, A054143.

Cf. A140513, A068565, A068566, A104709.

Sequence in context: A328464 A323956 A086272 * A110814 A275599 A210038

Adjacent sequences:  A104706 A104707 A104708 * A104710 A104711 A104712

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Mar 19 2005

STATUS

approved

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Last modified October 23 20:17 EDT 2019. Contains 328373 sequences. (Running on oeis4.)