A188464 Diagonal sums of triangle A188463.

1, 3, 8, 22, 62, 178, 519, 1533, 4578, 13800, 41937, 128345, 395232, 1223792, 3807903, 11900549, 37339043, 117574429, 371429284, 1176876762, 3739129185, 11909686261, 38022182028, 121648373964, 389979453010, 1252517211660, 4029754366713, 12986073134365

Offset: 0

Author

Paul Barry, Apr 01 2011

Keywords

• nonn
• easy

Comments

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)

Examples

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

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• S. Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020, see Theorem 2.1 for t=0 and r=1.

Crossrefs

Cf. A000108, A188463.

Programs

Magma

m:=30; R:=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) ));

Mathematica

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 *)

Maxima

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 */

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

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

Formula

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

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

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