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A023432
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Number of Dyck n-paths with ascents and descents of length equal to 1 (mod 3).
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10
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1, 1, 1, 1, 2, 4, 7, 12, 22, 42, 80, 152, 292, 568, 1112, 2185, 4313, 8557, 17050, 34089, 68370, 137542, 277475, 561185, 1137595, 2311014, 4704235, 9593662, 19598920, 40103635, 82185653, 168666493, 346613232, 713200114, 1469254621, 3030218948, 6256281188
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OFFSET
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0,5
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COMMENTS
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Number of Motzkin paths of length n-1 with no peaks, no double rises and no doubledescents (i.e., no UD's, no UU's and no DD's, where U=(1,1) and D=(1,-1), n>0; can be easily formulated using RNA secondary structure terminology). E.g., a(5)=4 because we have HHHH, HUHD, UHDH and UHHD; here H=(1,0). Also number of peakless Motzkin paths of length n in which each D=(1,-1) step is followed by an H=(1,0) step (can be easily formulated using RNA secondary structure terminology). E.g., a(5)=4 because we have HHHHH, HUHDH, UHDHH and UHHDH (here U=(1,1)). - Emeric Deutsch, Jan 09 2004
The coefficient of t^n in the power series expansion of the solution u in the equation (1-t*u)(u-t*u-t-t^2*u^2+t^3*u)=0. - Shanzhen Gao, May 13 2011
Also the number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 3). The a(5) = 4 paths for n=5 are: UDUDUDUDUD, UUUUDDDDUD, UUUUDUDDDD, UDUUUUDDDD. - Alois P. Heinz, May 09 2012
a(n)=number of strictly alternating bargraphs of semiperimeter n+2. A bargraph is said to be strictly alternating if its ascents and descents alternate and all the formed peaks and valleys have width 1. An ascent (descent) is a maximal sequence of consecutive U (D) steps. Example: a(4) = 2 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only those corresponding to the compositions [5] and [2,1,2] are strictly alternating. - Emeric Deutsch, Aug 26 2016
For n>=1, a(n) is the number of Dyck paths of semilength n+2 in which all ascent and descent lengths are >=3. For example, a(4) = 2 counts U^6.D^6, U^3.D^3.U^3.D^3 where ^ denotes repetition and a dot denotes concatenation. The gf F(x) = 1 + x^3 + x^4 + x^5 + 2*x^6 + ... for these paths satisfies F = 1 + x^3/(1-x) + (F-1)x^3/((1-x)(1-x*F)), which follows from a first return decomposition and summing over the lengths of the first ascent and first descent. A bijection to the title paths would be interesting. - David Callan, Dec 07 2021
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LINKS
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FORMULA
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G.f.: (1-z+z^3-sqrt(1-2z-2z^3+z^2-2z^4+z^6))/(2z^3). - Emeric Deutsch, Jan 09 2004
G.f.: 1/(1-x-x^4/(1-x-x^3-x^4/(1-x-x^3-x^4/(1-x-x^3-x^4/(1-... (continued fraction). - Paul Barry, May 22 2009
G.f.: 1/(1-x/(1-x^3/(1-x/(1-x^3/(1-x/(1-x^3/(1-... (continued fraction). - Paul Barry, Nov 30 2009
G.f. (for offset -1) satisfies: A(x) = (1 + x*A(x))*(1 + x^3*A(x)).
G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(2*k) ).
G.f.: A(x) = exp( Sum_{n>=1} x^n/n * (1-x^2)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2 * x^(2*k) ). (End)
a(n) ~ sqrt(3-5*r+2*r^2-3*r^3-2*r^4) / (2*sqrt(2*Pi)*n^(3/2)*r^(n+3)), where r = 0.465571231876768... is the root of the equation 1+r^2+r^6 = 2*r*(1+r^2+r^3). - Vaclav Kotesovec, Mar 22 2014
a(n) = Sum_{k=0..(n-1)/2}(C(n-2*k,k)*C(n-2*k,k+1)/(n-2*k), n>0, a(0)=1. - Vladimir Kruchinin, Jan 21 2019
D-finite with recurrence (n+3)*a(n) +(-2*n-3)*a(n-1) +n*a(n-2) +(-2*n+3)*a(n-3) +2*(-n+3)*a(n-4) +(n-6)*a(n-6)=0. - R. J. Mathar, Jul 23 2023
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 12*x^6 + 22*x^7 +...
where the logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^2)*x + (1 + 2^2*x^2 + x^4)*x^2/2 + (1 + 3^2*x^2 + 3^2*x^4 + x^6)*x^3/3 + (1 + 4^2*x^2 + 6^2*x^4 + 4^2*x^6 + x^8)*x^4/4 + (1 + 5^2*x^2 + 10^2*x^4 + 10^2*x^6 + 5^2*x^8 + x^10)*x^5/5 + ... - Paul D. Hanna
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MAPLE
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a:= proc(n) option remember;
`if`(n=0, 1, a(n-1) +add(a(k)*a(n-3-k), k=1..n-3))
end:
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MATHEMATICA
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Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-3-k ], {k, 0, n-4} ];
CoefficientList[Series[(1-x+x^3-Sqrt[1-2*x-2*x^3+x^2-2*x^4+x^6])/(2*x^3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 22 2014 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x^3*A +x*O(x^n))); polcoeff(A, n)} /* Paul D. Hanna */
(PARI) {a(n)=polcoeff( exp(sum(m=1, n+1, x^m/m*sum(j=0, m, binomial(m, j)^2*x^(2*j))+x*O(x^n))), n)} /* Paul D. Hanna */
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x^2)^(2*m+1)*sum(j=0, n\2, binomial(m+j, j)^2*x^(2*j))*x^m/m)+x*O(x^n))); polcoeff(A, n, x)} /* Paul D. Hanna */
(Haskell)
a023432 n = a023432_list !! n
a023432_list = 1 : 1 : f [1, 1] where
f xs'@(x:_:xs) = y : f (y : xs') where
y = x + sum (zipWith (*) xs $ reverse $ tail xs)
(Maxima)
a(n):=if n=0 then 1 else sum(binomial(n-2*q, q)*binomial(n-2*q, q+1)/(n-2*q), q, 0, (n-1)/2); /* Vladimir Kruchinin, Jan 21 2019 */
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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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EXTENSIONS
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STATUS
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approved
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