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A077587
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a(n) = C(n+1) + n*C(n) where C = A000108 (Catalan numbers).
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4
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1, 3, 9, 29, 98, 342, 1221, 4433, 16302, 60554, 226746, 854658, 3239044, 12332140, 47137005, 180780345, 695367510, 2681600130, 10364759790, 40142121030, 155748675420, 605274171060, 2355676013730, 9180275261274, 35819645937228
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OFFSET
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0,2
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COMMENTS
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Number of ascents of length 2 starting at an even level in all Dyck paths of semilength n+2. Example: a(1)=3 because all Dyck paths of semilength 3 are UDUDUD, UD(UU)DD, (UU)DDUD, (UU)DUDD and UUUDDD, where U=(1,1), D=(1,-1), having altogether 3 ascents of length 2 that start at an even level (shown between parentheses). - Emeric Deutsch, Nov 29 2005
a(n) is the number of parking functions of size n+1 avoiding the patterns 132, 231, and 321. - Lara Pudwell, Apr 10 2023
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LINKS
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FORMULA
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a(n) = binomial(2n+1, n+1) - binomial(2n, n+2).
a(n) = (3*(3*n+2)*a(n-1) - 2*(11*n-7)*a(n-2) + 4*(2*n-5)*a(n-3))/(n+2), n>2.
G.f.: A(x) = (1 - 3*x - (1-5*x+2*x^2)/sqrt(1-4*x) )/(2*x^2) satisfies 0 = (x^2+4*x-1) + (12*x^2-7*x+1)*A + (4*x^3-x^2)*A^2.
E.g.f.: A(x) = (1+x)B(x)' where B(x) = e.g.f. of A000108.
a(n) = Sum_{k=0..n} binomial(n,k)*A057977(k)*2^(n-k); here the A057977 are understood as the extended Catalan numbers (see also A063549). Related to Touchard's identity. - Peter Luschny, Jul 14 2016
Asymptotic starts a(n) ~ (4^n/sqrt(Pi*n))*(1 + (23/2^3)/n - (1199/2^7)/n^2 +(22685/2^10)/n^3 - (1562421/2^15)/n^4 + ... ). - Peter Luschny, Jul 14 2016
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MAPLE
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egf := x -> exp(2*x)*(1+1/x)*BesselI(1, 2*x);
seq(n!*coeff(series(egf(x), x, n+2), x, n), n=0..24); # Peter Luschny, Apr 14 2014
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MATHEMATICA
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Table[(CatalanNumber[n + 1] + n CatalanNumber[n]), {n, 0, 40}] (* Vincenzo Librandi, Apr 15 2014 *)
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PROG
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(PARI) a(n)=if(n<0, 0, (n^2+6*n+2)*(2*n)!/n!/(n+2)!)
(PARI) a(n)=if(n<0, 0, polcoeff((4+x+1/x-(x+1/x)^2)*(1+x)^(2*n), n)/2)
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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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