Search: id:A059345
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%I A059345
%S A059345 1,1,4,9,29,82,255,773,2410,7499,23575,74298,235325,747407,2381126,
%T A059345 7603433,24332595,78013192,250540055,805803691,2595158718,8368026845,
%U A059345 27012184877,87283372610,282294378071,913775677281,2960160734818
%N A059345 Central column of Pascal's "rhombus" (actually a triangle) A059317.
%C A059345 Number of paths in the right half-plane from (0,0) to (n,0) consisting 
               of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(3)=9 
               because we have hhh, hH, Hh, hUD, hDU, UhD, DhU, UDh and DUh. The 
               number of such paths restricted to the first quadrant is given in 
               A128720. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
%C A059345 Other two columns of the triangle in A059317 are given in A106053 and 
               A106050. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
%D A059345 J. Goldwasser et al., The density of ones in Pascal's rhombus, Discrete 
               Math., 204 (1999), 231-236.
%H A059345 T. D. Noe, <a href="b059345.txt">Table of n, a(n) for n=0..200</a>
%F A059345 G.f.=1/sqrt[(1+z-z^2)(1-3z-z^2)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Sep 03 2007
%F A059345 Rec. rel.: (n+1)a(n+1)=(2n+1)a(n)+5na(n-1)-(2n-1)a(n-2)-(n-1)a(n-3). 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
%F A059345 a(n)=sum{k=0..floor(n/2), C(n-k,k)*A002426(n-2k)}; [From Paul Barry (pbarry(AT)wit.ie), 
               Nov 29 2008]
%p A059345 r:=proc(i,j) if i=0 then 0 elif i=1 and abs(j)>0 then 0 elif i=1 and 
               j=0 then 1 elif i>=1 then r(i-1,j)+r(i-1,j-1)+r(i-1,j+1)+r(i-2,j) 
               else 0 fi end: seq(r(i,0),i=1..12); # very slow (from Emeric Deutsch, 
               Jun 06 2004)
%p A059345 G:=1/sqrt((1+z-z^2)*(1-3*z-z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,
               z, n),n=0..27); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 
               03 2007
%p A059345 a[0]:=1: a[1]:=1: a[2]:=4: a[3]:=9: for n from 3 to 26 do a[n+1]:=((2*n+1)*a[n]+5*n*a[n-1]-(2*n-1)*a[n-2]-(n-\
               1)*a[n-3])/(n+1) end do: seq(a[n],n=0..27); - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Sep 03 2007
%Y A059345 Cf. A128720, A106050, A106053.
%Y A059345 Sequence in context: A000368 A094255 A069563 this_sequence A127768 A091658 
               A086688
%Y A059345 Adjacent sequences: A059342 A059343 A059344 this_sequence A059346 A059347 
               A059348
%K A059345 nonn,easy,nice
%O A059345 0,3
%A A059345 N. J. A. Sloane (njas(AT)research.att.com), Jan 27 2001
%E A059345 More terms from Larry Reeves (larryr(AT)acm.org), Jan 30 2001

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