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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)).

%I A143950 #9 Jul 21 2017 10:47:37
%S A143950 1,1,1,1,2,3,5,7,2,12,20,10,30,61,36,5,79,182,133,35,213,547,488,168,
%T A143950 14,584,1668,1728,756,126,1628,5116,6020,3240,750,42,4600,15752,20812,
%U A143950 13200,3960,462,13138,48709,71376,52030,19360,3267,132,37871,151164
%N 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)).
%C A143950 Row n contains 1 + floor(n/2) entries.
%C A143950 Row sums are the Catalan numbers (A000108).
%C A143950 T(n,0) = A101785(n).
%C A143950 Sum_{k=0..floor(n/2)} k*T(n,k) = A014301(n).
%C A143950 For the Dyck path statistic "number of odd-length ascents" see A096793.
%F A143950 G.f. G=G(s,z) satisfies G = 1 + zG(1 + szG)/(1 - z^2*G^2).
%F A143950 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).
%e A143950 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).
%e A143950 Triangle starts:
%e A143950    1;
%e A143950    1;
%e A143950    1,  1;
%e A143950    2,  3;
%e A143950    5,  7,  2;
%e A143950   12, 20, 10;
%e A143950   30, 61, 36,  5;
%p A143950 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
%Y A143950 Cf. A000108, A014301, A096793, A101785.
%K A143950 nonn,tabf
%O A143950 0,5
%A A143950 _Emeric Deutsch_, Oct 05 2008