Search: id:A132276
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%I A132276
%S A132276 1,1,1,3,2,1,6,7,3,1,16,18,12,4,1,40,53,37,18,5,1,109,148,120,64,25,6,
               1,
%T A132276 297,430,369,227,100,33,7,1,836,1244,1146,760,385,146,42,8,1,2377,3656,
%U A132276 3519,2518,1391,606,203,52,9,1,6869,10796,10839,8188,4900,2346,903,272
%N A132276 Triangle read by rows: T(n,k) is the number of paths in the first quadrant 
               from (0,0) to (n,k), consisting of steps U=(1,1), D=(1,-1), h=(1,
               0) and H=(2,0) (0<=k<=n).
%C A132276 T(n,0)=A128720(n).
%C A132276 Mirror image of A059397. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 18 2007
%C A132276 Row sums yield A059398.
%D A132276 W. Klostermeyer et al., A Pascal rhombus, Fibonacci Quarterly, 35 (1976), 
               318-328.
%D A132276 W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, 
               Fibonacci Quarterly, 35 (1997), 318-328.
%F A132276 G.f.=G(t,z)=g/(1-tzg), where g=1+zg+z^2*g+z^2*g^2 or g=c(z^2/(1-z-z^2)^2)/
               (1-z-z^2), where c=[(1-sqrt(1-4z)]/(2z) is the Catalan function.
%F A132276 T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-1,k+1)+T(n-2,k). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 18 2007
%F A132276 Column k has g.f. z^k*g^(k+1), where g=1+zg+z^2*g+z^2*g^2=[1-z-z^2-sqrt((1+z-z^2)(1-3z--z^2))]/
               (2z^2).
%e A132276 Triangle begins:
%e A132276 1,
%e A132276 1,1,
%e A132276 3,2,1,
%e A132276 6,7,3,1,
%e A132276 16,18,12,4,1,
%e A132276 40,53,37,18,5,1,
%e A132276 109,148,120,64,25,6,1,
%e A132276 T(3,2)=3 because we have UUh, UhU and hUU.
%p A132276 g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: G:=simplify(g/(1-t*z*g)): 
               Gser:=simplify(series(G,z=0,13)): for n from 0 to 10 do P[n]:=sort(coeff(Gser, 
               z,n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, j), j = 0 
               .. n) end do; # yields sequence in triangular form
%Y A132276 Cf. A059397, A128720 (the leading diagonal).
%Y A132276 Cf. A059398.
%Y A132276 Sequence in context: A114155 A079513 A139624 this_sequence A114586 A052174 
               A111049
%Y A132276 Adjacent sequences: A132273 A132274 A132275 this_sequence A132277 A132278 
               A132279
%K A132276 nonn,tabl
%O A132276 0,4
%A A132276 Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 16 2007, Sep 03 2007

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