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A133084 A007318 * A133080. 6
1, 2, 1, 3, 2, 1, 4, 3, 4, 1, 5, 4, 10, 4, 1, 6, 5, 20, 10, 6, 1, 7, 6, 35, 20, 21, 6, 1, 8, 7, 56, 35, 56, 21, 8, 1, 9, 8, 84, 56, 126, 56, 36, 8, 1, 10, 9, 120, 84, 252, 126, 120, 36, 10, 1, 11, 10, 165, 120, 462, 252, 330, 120, 55, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums = A003945: (1, 3, 6, 12, 24, 48, 96,...).

A133084 is jointly generated with A133567 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+v(n-1,x)+1.  See the Mathematica section. - Clark Kimberling, Feb 28 2012

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

Binomial transform of triangle A133080.

EXAMPLE

First few rows of the triangle are:

1;

2, 1;

3, 2, 1;

4, 3, 4, 1;

5, 4, 10, 4, 1;

6, 5, 20, 10, 6, 1;

7, 6, 35, 20, 21, 6, 1;

...

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]   (* A133567 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]   (* A133084 *)

(* Clark Kimberling, Feb 28 2012 *)

T[n_, k_] := If[k == n, 1, (1  - (1 + (-1)^k)/2 )*Binomial[n, k] + ((1 + (-1)^k)/2)*Binomial[n - 1, k - 1]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

PROG

(PARI) for(n=1, 10, for(k=1, n, print1(if(k == n, 1, (1  - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)), ", "))) \\ G. C. Greubel, Oct 21 2017

(MAGMA) /* As triangle */ [[(1-(1+(-1)^k)/2 )*Binomial(n, k)+((1+(-1)^k)/2)*Binomial(n-1, k-1): k in [1..n]]: n in [1.. 11]]; // Vincenzo Librandi, Oct 21 2017

CROSSREFS

Cf. A133080, A003945, A133567.

Sequence in context: A135227 A104325 A204925 * A118851 A112383 A272464

Adjacent sequences:  A133081 A133082 A133083 * A133085 A133086 A133087

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Sep 16 2007

STATUS

approved

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Last modified February 19 07:39 EST 2018. Contains 299330 sequences. (Running on oeis4.)