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 A160232 Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n. 11

%I

%S 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,

%T 7,27,70,129,168,144,64,1,8,35,104,225,360,416,320,128,1,9,44,147,363,

%U 681,968,1008,704,256,1,10,54,200,553,1182,1970,2528,2400,1536,512,1,11,65

%N Array read by antidiagonals: row n has g.f. ((1-x)/(1-2x))^n.

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

%C As triangle, mirror image of A105306. - From _Philippe Deléham_, Nov 01 2011

%C 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.

%C [From Clark Kimberling, Feb 25 2012]

%C 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

%F Contribution from _Philippe Deléham_, Mar 08 2012. (Start)

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

%F 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.

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

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

%e Array begins:

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

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

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

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

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

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

%e As a triangle (see Comments):

%e 1

%e 1 1

%e 1 2 2

%e 1 3 5 4

%e 1 4 9 12 8

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

%e 1

%e 1, 0

%e 1, 1, 0

%e 1, 2, 2, 0

%e 1, 3, 5, 4, 0

%e 1, 4, 9, 12, 8, 0

%e 1, 5, 14, 25, 28, 16, 0

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

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

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

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

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

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

%t TableForm[cu]

%t Flatten[%] (* A160232 *)

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

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

%t TableForm[cv]

%t Flatten[%] (* A208341 *)

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

%Y Cf. A062110, A105306, A208341.

%K nonn,tabl

%O 1,5

%A _N. J. A. Sloane_, May 15 2010

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