A191796 Number of DUU's in all length n left factors of Dyck paths; here U=(1,1) and D=(1,-1).
0, 0, 0, 0, 1, 3, 9, 21, 52, 113, 261, 550, 1226, 2542, 5546, 11389, 24494, 49989, 106413, 216258, 456826, 925586, 1943550, 3929090, 8210896, 16571018, 34494114, 69523116, 144246532, 290424604, 600907508, 1208835421, 2495229602, 5016122029, 10332784253, 20759855626
Offset: 0
Keywords
Examples
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).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Crossrefs
Cf. A191795.
Programs
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Maple
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);
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Mathematica
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 *) -
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
Formula
a(n) = Sum_{k>=0} k*A191795(n,k).
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)).
a(n) ~ 2^(n-5/2)*sqrt(n)/sqrt(Pi) * (1 + sqrt(Pi)/sqrt(2*n)). - Vaclav Kotesovec, Mar 21 2014
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