A143950 Triangle read by rows: T(n,k) is the number of Dyck n-paths containing k even-length ascents (0 <= k <= floor(n/2)).
1, 1, 1, 1, 2, 3, 5, 7, 2, 12, 20, 10, 30, 61, 36, 5, 79, 182, 133, 35, 213, 547, 488, 168, 14, 584, 1668, 1728, 756, 126, 1628, 5116, 6020, 3240, 750, 42, 4600, 15752, 20812, 13200, 3960, 462, 13138, 48709, 71376, 52030, 19360, 3267, 132, 37871, 151164
Offset: 0
Examples
T(4,1)=7 because we have UDUD(UU)DD, UD(UU)DDUD, UD(UU)DUDD, (UU)DDUDUD, (UU)DUDDUD, (UU)DUDUDD and (UUUU)DDDD (the even-length ascents are shown between parentheses). Triangle starts: 1; 1; 1, 1; 2, 3; 5, 7, 2; 12, 20, 10; 30, 61, 36, 5;
Programs
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Maple
eq:=G=1+(1+s*z*G)*z*G/(1-z^2*G^2): G:=RootOf(eq,G): Gser:=simplify(series(G,z =0,16)): for n from 0 to 13 do P[n]:=sort(expand(coeff(Gser,z,n))) end do: for n from 0 to 13 do seq(coeff(P[n],s,j),j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
Formula
G.f. G=G(s,z) satisfies G = 1 + zG(1 + szG)/(1 - z^2*G^2).
The trivariate g.f. H=H(t,s,z), where t (s) marks odd-length (even-length) ascents satisfies H = 1 + zH(t+szH)/(1-z^2*H^2).
Comments