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Number of peaks at odd level in all Dyck paths of semilength n that have no ascents and no descents of length 1.
3

%I #15 Jul 26 2022 13:31:40

%S 0,0,0,1,0,5,4,23,36,123,252,720,1664,4427,10804,27971,69972,179469,

%T 454300,1162529,2961056,7579620,19376728,49659406,127263208,326610827,

%U 838550920,2154985059,5540935616,14257159799,36703613556,94544579575

%N Number of peaks at odd level in all Dyck paths of semilength n that have no ascents and no descents of length 1.

%H G. C. Greubel, <a href="/A167636/b167636.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} k*A167634(n,k).

%F G.f.: G(z) = z(1 - z^2 - 2z^3 + z^4 - (1 + z - z^2)*sqrt((1 + z + z^2)(1 - 3z + z^2)))/(2(1 + z - z^2)sqrt((1 + z + z^2)(1 - 3z + z^2))).

%F a(n) ~ sqrt(3/sqrt(5)-1) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+5/2)). - _Vaclav Kotesovec_, Mar 20 2014

%F Equivalently, a(n) ~ phi^(2*n - 1) / (4 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 08 2021

%F D-finite with recurrence +(n-1)*(38*n-213)*a(n) +38*(n-2)*(n-4)*a(n-1) +4*(-84*n^2+680*n-1185)*a(n-2) -26*(9*n-19)*(n-4)*a(n-3) +(n-5)*(356*n-1879)*a(n-4) +2*(111*n-491)*(n-6)*a(n-5) +2*(95*n-137)*(n-7)*a(n-6) -50*(8*n-23)*(n-8)*a(n-7) +3*(36*n-97)*(n-9)*a(n-8)=0. - _R. J. Mathar_, Jul 26 2022

%e a(5)=5 because UUDDUU(UD)DD, UU(UD)DDUUDD, UU(UD)DU(UD)DD, and UUUU(UD)DDDD have 1 + 1 + 2 + 1 = 5 odd-level peaks (shown between parentheses).

%p G := (1/2)*z*(1-z^2-2*z^3+z^4-(1+z-z^2)*sqrt((1+z+z^2)*(1-3*z+z^2)))/((1+z-z^2)*sqrt((1+z+z^2)*(1-3*z+z^2))): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);

%t CoefficientList[Series[1/2*x*(1-x^2-2*x^3+x^4-(1+x-x^2)*Sqrt[(1+x+x^2)*(1-3*x+x^2)])/((1+x-x^2)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)

%o (PARI) x='x+O('x^50); concat([0,0,0], Vec(1/2*x*(1-x^2-2*x^3+x^4-(1+x-x^2)*sqrt((1+x+x^2)*(1-3*x+x^2)))/((1+x-x^2)*sqrt((1+x+x^2)*(1-3*x+x^2))))) \\ _G. C. Greubel_, Feb 12 2017

%Y Cf. A167634, A167639.

%K nonn

%O 0,6

%A _Emeric Deutsch_, Nov 08 2009