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%I #25 Jan 15 2018 02:50:11
%S 1,1,2,2,3,4,3,8,8,8,5,15,25,20,16,8,30,55,70,48,32,13,56,125,175,184,
%T 112,64,21,104,262,440,512,464,256,128,34,189,539,1014,1401,1416,1136,
%U 576,256,55,340,1075,2270,3501,4170,3760,2720,1280,512
%N Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.
%C A164975 is jointly generated with A209125 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)=u(n-1,x)+ 2x*v(n-1,x). See the Mathematica section. - _Clark Kimberling_, Mar 05 2012
%H G. C. Greubel, <a href="/A164975/b164975.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F T(n,n-1) = A000079(n-1).
%F T(n,n-2) = A001792(n-2). - _R. J. Mathar_, Jan 27 2011
%F T(n,1) = A099920(n-1). - _R. J. Mathar_, Jan 27 2011
%F G.f.: x/(1-2*y*x-x-x^2+y*x^2). - _Philippe Deléham_, Mar 21 2012
%F Sum_{k=0..n-1, n>0} T(n,k)*x^k = A000045(n), A000244(n-1), A004254(n), A186446(n-1), A190980(n) for x = 0, 1, 2, 3, 4 respectively. - _Philippe Deléham_, Mar 21 2012
%e Triangle T(n,k), 0 <= k < n, n >= 1, begins:
%e 1;
%e 1, 2;
%e 2, 3, 4;
%e 3, 8, 8, 8;
%e 5, 15, 25, 20, 16;
%e 8, 30, 55, 70, 48, 32;
%e 13, 56, 125, 175, 184, 112, 64;
%e 21, 104, 262, 440, 512, 464, 256, 128;
%e ...
%e T(7,1) = 30 + 2*8 + 15 - 5 = 56.
%e T(6,1) = 15 + 2*5 + 8 - 3 = 30.
%p A164975 := proc(n,k) option remember; if n <=0 or k > n or k< 1 then 0; elif k= 1 then combinat[fibonacci](n); else procname(n-1,k)+2*procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc: # _R. J. Mathar_, Jan 27 2011
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + (x + 1)*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[%] (* A209125 *)
%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[%] (* A164975 *)
%t (* _Clark Kimberling_, Mar 05 2012 *)
%t With[{nmax = 10}, Rest[CoefficientList[CoefficientList[Series[ x/(1 - 2*y*x-x-x^2+y*x^2), {x,0,nmax}, {y,0,nmax}], x], y]]//Flatten] (* _G. C. Greubel_, Jan 14 2018 *)
%Y Cf. A000045, A000079, A000244 (row sums).
%K nonn,tabl
%O 1,3
%A _Mark Dols_, Sep 03 2009
%E Corrected by _Philippe Deléham_, Mar 21 2012
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