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%I #23 Jan 27 2020 01:46:47
%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. - _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. - _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 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..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 From _Clark Kimberling_, Feb 25 2012: (Start)
%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; (End)
%e From _Philippe Deléham_, Mar 08 2012: (Start)
%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; (End)
%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 *)
%t (* _Clark Kimberling_, Feb 25 2012 *)
%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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