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%I #11 Mar 31 2015 03:24:18
%S 1,1,1,1,2,3,6,7,1,18,19,5,54,59,18,1,166,191,65,7,522,631,242,34,1,
%T 1670,2123,906,154,9,5418,7247,3395,680,55,1,17786,25011,12746,2932,
%U 300,11,58974,87071,47931,12414,1540,81,1,197226,305275,180439,51878,7552
%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k ascents of length 2 starting at an even level (0<=k<=floor(n/2)).
%C Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). Sum(kT(n,k), k=0..floor(n/2)) = binomial(2n-3,n-1)-binomial(2n-4,n) = A077587(n-2) (n>=2). Column 0 yields A114464.
%H Alois P. Heinz, <a href="/A114462/b114462.txt">Rows n = 0..200, flattened</a>
%F G.f.: G(t,z) satisfies zG^2-(1-z+tz-3tz^2+3z^2-z^3-t^2z^3+2tz^3)G+1-z+z^2+tz-tz^2=0.
%e T(4,1) = 7 because we have (UU)DDUDUD, UD(UU)DDUD, UDUD(UU)DD, (UU)DUDDUD,
%e UD(UU)DUDD, (UU)DUDUDD and (UU)DUUDDD, where U=(1,1), D=(1,-1) (the ascents of length 2 starting at an even level are shown between parentheses; note that the last path has an ascent of length 2 that starts at an odd level).
%e Triangle starts:
%e 1;
%e 1;
%e 1, 1;
%e 2, 3;
%e 6, 7, 1;
%e 18, 19, 5;
%e 54, 59, 18, 1;
%p G:= 1/2/z*(3*z^2+2*z^3*t+1-z^3*t^2-3*z^2*t-z^3+t*z-z -sqrt(1+20*z^3*t-18*z^5*t^2+15*z^4*t^2+18*z^5*t+6*z^5*t^3-2*z^4*t^3-12*z^2*t -12*z^3 -6*z-24*z^4*t-8*z^3*t^2+z^6-6*z^5+11*z^4 +z^2*t^2+6*z^6*t^2 -4*z^6*t^3 -4*z^6*t+z^6*t^4+2*t*z +11*z^2)): Gser:=simplify(series(G,z=0,17)): P[0]:=1: for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 14 do seq(coeff(t*P[n],t^j),j=1..1+floor(n/2)) od; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0,
%p `if`(t=2, z, 1), expand(b(x-1, y-1, min(3, t+1))+
%p `if`(t=2 and irem(y, 2)=0, z, 1)*b(x-1, y+1, 0))))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0$2)):
%p seq(T(n), n=0..14); # _Alois P. Heinz_, Mar 12 2014
%t b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, If[t==2, z, 1], Expand[ b[x-1, y-1, Min[3, t+1]] + If[t==2 && Mod[y, 2]==0, z, 1]*b[x-1, y+1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Mar 31 2015, after _Alois P. Heinz_ *)
%Y Cf. A077587, A000108, A114463, A114464, A114465, A102402.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Nov 29 2005
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