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A023432
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Generalized Catalan Numbers.
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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 a 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 (deutsch(AT)duke.poly.edu), 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 [From shanzhen Gao, May 13,2011]
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REFERENCES
| S.Gao, H.Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).
M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problemes d'enumeration en biologie moleculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
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LINKS
| M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux at problemes d'enumeration en biologie moleculaire, Sem. Loth. Comb. B08l (1984) 79-86.
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FORMULA
| G.f.=[1-z+z^3-sqrt(1-2z-2z^3+z^2-2z^4+z^6)]/(2z^3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 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). [From Paul Barry (pbarry(AT)wit.ie), May 22 2009]
G.f.: 1/(1-x/(1-x^3/(1-x/(1-x^3/(1-x/(1-x^3/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Nov 30 2009]
G.f.: (1-t+t^3-sqrt((1-t-t^3)^2-4∗t^4)))/(2∗t^2) [From Shanzhen Gao, May 13,2011]
Contribution from Paul D. Hanna, Nov 01 2011: (Start)
G.f. 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)
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EXAMPLE
| 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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MATHEMATICA
| Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-3-k ], {k, 0, n-4} ];
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PROG
| (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 */
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CROSSREFS
| Cf. A000108, A001006, A004148, A006318.
Sequence in context: A018179 A190165 A127542 * A072641 A135360 A082548
Adjacent sequences: A023429 A023430 A023431 * A023433 A023434 A023435
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KEYWORD
| nonn,easy
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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EXTENSIONS
| More terms after term 30 from Paul Barry (pbarry(AT)wit.ie), Nov 30 2009
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