cp's OEIS Frontend

A191796 Number of DUU's in all length n left factors of Dyck paths; here U=(1,1) and D=(1,-1).

%I A191796 #13 Mar 28 2017 15:02:42
%S A191796 0,0,0,0,1,3,9,21,52,113,261,550,1226,2542,5546,11389,24494,49989,
%T A191796 106413,216258,456826,925586,1943550,3929090,8210896,16571018,
%U A191796 34494114,69523116,144246532,290424604,600907508,1208835421,2495229602,5016122029,10332784253,20759855626
%N A191796 Number of DUU's in all length n left factors of Dyck paths; here U=(1,1) and D=(1,-1).
%H A191796 G. C. Greubel, <a href="/A191796/b191796.txt">Table of n, a(n) for n = 0..1000</a>
%F A191796 a(n) = Sum_{k>=0} k*A191795(n,k).
%F A191796 G.f.: ((1-3*z^2-z^3)*sqrt(1-4*z^2) -1+5*z^2+z^3-4*z^4)/(2*z*(1-2*z)*sqrt(1-4*z^2)).
%F A191796 a(n) ~ 2^(n-5/2)*sqrt(n)/sqrt(Pi) * (1 + sqrt(Pi)/sqrt(2*n)). - _Vaclav Kotesovec_, Mar 21 2014
%F A191796 Conjecture: -(n+1)*(2527*n^2+15963*n-146560)*a(n) +(-2527*n^3+68000*n^2-231053*n-293120)*a(n-1) +2*(12635*n^3+906*n^2-429395*n+746484)*a(n-2) +4*(2527*n^3-70527*n^2+316742*n-316524)*a(n-3) -24*(n-5)*(2527*n^2-232*n-28664)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016
%e A191796 a(4)=1 because in UDUD, U(DUU), UUDD, UUDU, UUUD, and UUUU the  total number of DUUs is 0 + 1 + 0 + 0 +0 + 0 = 1 (shown between parentheses).
%p A191796 g := (((1-3*z^2-z^3)*sqrt(1-4*z^2)-1+5*z^2+z^3-4*z^4)*1/2)/(z*(1-2*z)*sqrt(1-4*z^2)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
%t A191796 CoefficientList[Series[(((1-3*x^2-x^3)*Sqrt[1-4*x^2]-1+5*x^2+x^3-4*x^4)/2) / (x*(1-2*x)*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)
%o A191796 (PARI) z='z+O('z^50); concat([0,0,0,0], Vec(((1-3*z^2-z^3)*sqrt(1-4*z^2) -1+5*z^2+z^3-4*z^4)/(2*z*(1-2*z)*sqrt(1-4*z^2)))) \\ _G. C. Greubel_, Mar 28 2017
%Y A191796 Cf. A191795.
%K A191796 nonn
%O A191796 0,6
%A A191796 _Emeric Deutsch_, Jun 18 2011