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A138164
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Row sums of Riordan array (c(-x^2),xc(-x^2)^2)^(-1) where c(x) is the g.f. of A000108.
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1
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1, 1, 2, 4, 9, 20, 47, 109, 262, 622, 1516, 3653, 8988, 21883, 54213, 133004, 331233, 817432, 2044151, 5068346, 12716872, 31651555, 79636493, 198843284, 501466519, 1255489165, 3172569392, 7961388439, 20152910577, 50674576772
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OFFSET
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0,3
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COMMENTS
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A transform of the F(n+1) by (1,x(1-x^2))^(-1).
(c(-x^2),xc(-x^2)^2)^(-1) factors as (1,x(1-x^2))^(-1)*(1/(1-x^2),x/(1-x^2)).
It appears that a(n) is the number of Dyck paths (A000108) of semilength n in which all non-return descents are of even length (a return descent is a maximal sequence of downsteps that returns the path to ground level). For example, a(4) = 9 counts, among others, UUUDDUDD and UDUUUDDD but not UUDUUDDD. - David Callan, Nov 13 2021
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LINKS
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FORMULA
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G.f.: 1/(1-v-v^2) where v=(2/sqrt(3))*sin(arcsin(x*3*sqrt(3)/2)/3) is the series reversion of x(1-x^2). [Corrected by Paul D. Hanna, Feb 24 2015]
Let M = the production matrix:
1, 1, 0, 0, 0, 0, 0, 0, ...
1, 0, 1, 0, 0, 0, 0, 0, ...
1, 1, 0, 1, 0, 0, 0, 0, ...
1, 0, 1, 0, 1, 0, 0, 0, ...
1, 1, 0, 1, 0, 1, 0, 0, ...
1, 0, 1, 0, 1, 0, 1, 0, ...
1, 1, 0, 1, 0, 1, 0, 1, ...
...
a(n) = top left term of M^n. a(n+1) = sum of top row terms of M^n. Example: top row of M^3 = (4, 3, 1, 1), where a(3) = 4 and a(4) = 9 = (4 + 3 + 1 + 1). (End)
Conjecture: -8*n*(n-1)*a(n) + 12*(n+3)*(n-1)*a(n-1) + 2*(89*n^2-512*n+651)*a(n-2) + 3*(-127*n^2+715*n-966)*a(n-3) + 3*(-247*n^2+2431*n-5698)*a(n-4) + 9*(75*n-341)*(3*n-16)*a(n-5) - 72*(3*n-14)*(3*n-16)*a(n-6) = 0. - R. J. Mathar, Feb 24 2015
Recurrence (of order 4): 4*(n-1)*n*(11*n^2 - 50*n + 48)*a(n) = 12*(n-1)*(11*n^3 - 50*n^2 + 66*n - 40)*a(n-1) + (253*n^4 - 2294*n^3 + 6379*n^2 - 5898*n + 720)*a(n-2) - 3*(297*n^4 - 2538*n^3 + 7347*n^2 - 7946*n + 2000)*a(n-3) + 3*(3*n - 10)*(3*n - 8)*(11*n^2 - 28*n + 9)*a(n-4).
a(n) ~ (45*(1 - (-1)^n) + 26*sqrt(3)*(1 + (-1)^n)) * 3^(3*n/2) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). (End)
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MAPLE
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vx := 2/sqrt(3)*sin(arcsin(x*3*sqrt(3)/2)/3) ;
1/(1-vx-vx^2) ;
coeftayl(%, x=0, n) ;
subs(4^(1/2)=2, %) ;
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MATHEMATICA
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CoefficientList[Series[1/(1/3 - 2*Sin[1/3*ArcSin[3*Sqrt[3]*x/2]]/Sqrt[3] + 2*Cos[2/3*ArcSin[3*Sqrt[3]*x/2]]/3), {x, 0, 30}], x] (* Vaclav Kotesovec, Nov 15 2021 *)
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PROG
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(PARI) my(x='x+O('x^66)); v=serreverse(x*(1-x^2)); Vec(1/(1-v-v^2)) \\ Joerg Arndt, Feb 24 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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