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A188464
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Diagonal sums of triangle A188463.
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4
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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
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OFFSET
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0,2
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COMMENTS
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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)
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REFERENCES
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S. Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, Discrete Math., 344 (2021), no. 6, 112364.
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LINKS
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FORMULA
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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..(n+2)/2} C(2*m,m)/(m+1)*C(n+m+1,3*m-1)). - Vladimir Kruchinin, Jan 24 2022
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EXAMPLE
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For n=1, Dyck 4-paths are (2,-1,2,-3), (3,-1,1,-3) and (3,-2,1,-2), a(1) = 3.
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MATHEMATICA
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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 *)
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PROG
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(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
(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) ));
(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
(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 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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