login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A160232 Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n. 12
1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 8, 1, 5, 14, 25, 28, 16, 1, 6, 20, 44, 66, 64, 32, 1, 7, 27, 70, 129, 168, 144, 64, 1, 8, 35, 104, 225, 360, 416, 320, 128, 1, 9, 44, 147, 363, 681, 968, 1008, 704, 256, 1, 10, 54, 200, 553, 1182, 1970, 2528, 2400, 1536, 512, 1, 11, 65 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Suggested by a question from Phyllis Chinn (Humboldt State University).

As triangle, mirror image of A105306. - Philippe Deléham, Nov 01 2011

A160232 is jointly generated with A208341 as a triangular array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n > 1, u(n,x) = u(n-1,x) + x*v(n-1)x and v(n,x) = u(n-1,x) + 2x*v(n-1,x).  See the Mathematica section. - Clark Kimberling, Feb 25 2012

Subtriangle of the triangle T(n,k) given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012

LINKS

Table of n, a(n) for n=1..69.

FORMULA

From Philippe Deléham, Mar 08 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k < 0 or if k > n.

G.f.: (1-2*y*x)/(1-2*y*x-x+y*x^2).

Sum_{k=0..n, n>0} T(n,k)*x^k = A000012(n), A001519(n), A052984(n-1) for x = 0, 1, 2 respectively. (End)

EXAMPLE

Array begins:

  1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, ...

  1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, ...

  1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, 13056, 29952, 68096, 153600, 344064, 765952, 1695744, 3735552, ...

  1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, ...

  1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, 102592, 261760, 657920, 1632000, 4001280, 9708544, ...

  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, 236640, 632448, 1661056, 4296192, 10961664, 27630592, ...

From Clark Kimberling, Feb 25 2012: (Start)

As a triangle (see Comments):

  1;

  1,  1;

  1,  2,  2;

  1,  3,  5,  4;

  1,  4,  9, 12,  8;  (End)

From Philippe Deléham, Mar 08 2012: (Start)

(1, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins:

  1;

  1,  0;

  1,  1,  0;

  1,  2,  2,  0;

  1,  3,  5,  4,  0;

  1,  4,  9, 12,  8,  0;

  1,  5, 14, 25, 28, 16,  0; (End)

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

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

v[n_, x_] := u[n - 1, x] + 2*x*v[n - 1, x];

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[%]  (* A160232 *)

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

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

TableForm[cv]

Flatten[%]  (* A208341 *)

(* Clark Kimberling, Feb 25 2012 *)

CROSSREFS

Rows give A011782, A045623, A058396, A062109, A169792-A169797.

Cf. A062110, A105306, A208341.

Sequence in context: A079956 A140717 A257005 * A026300 A099514 A228352

Adjacent sequences:  A160229 A160230 A160231 * A160233 A160234 A160235

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, May 15 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 21 01:15 EST 2021. Contains 340332 sequences. (Running on oeis4.)