A188464 - OEIS

%I #35 Oct 19 2024 21:41:22

%S 1,3,8,22,62,178,519,1533,4578,13800,41937,128345,395232,1223792,

%T 3807903,11900549,37339043,117574429,371429284,1176876762,3739129185,

%U 11909686261,38022182028,121648373964,389979453010,1252517211660,4029754366713,12986073134365

%N Diagonal sums of triangle A188463.

%C Apparently, number of Dyck (n+3)-paths with no descent having the same length as the preceding ascent. - _David Scambler_, Apr 28 2012 (Proved by S. Elizalde, Disc. Math., 2021)

%D S. Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, Discrete Math., 344 (2021), no. 6, 112364.

%H G. C. Greubel, <a href="/A188464/b188464.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Elizalde, <a href="https://arxiv.org/abs/2008.05669">Symmetric peaks and symmetric valleys in Dyck paths</a>, arXiv:2008.05669 [math.CO], 2020, see Theorem 2.1 for t=0 and r=1.

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

%F Conjecture: (n+4)*a(n)-(4*n+9)*a(n-1) +(2*n-1)*a(n-2) -a(n-3) +(n-3)*a(n-4)=0. - _R. J. Mathar_, Nov 17 2011

%F a(n) = Sum_{m=1..floor((n+2)/2)} C(2*m,m)/(m+1)*C(n+m+1,3*m-1). - _Vladimir Kruchinin_, Jan 24 2022

%e For n=1, Dyck 4-paths are (2,-1,2,-3), (3,-1,1,-3) and (3,-2,1,-2), a(1) = 3.

%t CoefficientList[Series[(1-3*x+x^2-x^3-(1-x)*Sqrt[1-4*x+2*x^2+x^4])/( 2*x^4), {x,0,30}], x] (* _G. C. Greubel_, Nov 16 2018 *)

%o (PARI) x='x+O('x^30); Vec((1-3*x+x^2-x^3-(1-x)*sqrt(1-4*x+2*x^2 +x^4))/( 2*x^4)) \\ _G. C. Greubel_, Nov 16 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-3*x+x^2-x^3-(1-x)*Sqrt(1-4*x+2*x^2 +x^4))/( 2*x^4) ));

%o (Sage) s=((1-3*x+x^2-x^3-(1-x)*sqrt(1-4*x+2*x^2+x^4))/( 2*x^4)).series(x, 30); s.coefficients(x, sparse=False) # _G. C. Greubel_, Nov 16 2018

%o (Maxima)

%o a(n):=sum((binomial(2*m,m)*binomial(n+m+1,3*m-1))/(m+1),m,1,(n+2)/2); /* _Vladimir Kruchinin_, Jan 24 2022 */

%Y Cf. A000108, A188463.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Apr 01 2011