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A062980
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a(0) = 1, a(1) = 5; for n>1, a(n) = 6n*a(n-1) + Sum_{k=1..n-2} a(k)*a(n-k-1).
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3
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1, 5, 60, 1105, 27120, 828250, 30220800, 1282031525, 61999046400, 3366961243750, 202903221120000, 13437880555850250, 970217083619328000, 75849500508999712500, 6383483988812390400000, 575440151532675686278125
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of rooted unlabeled connected triangular maps on a compact closed oriented surface with 2n faces (and thus 3n edges). [Vidal]
Equivalently, the number of pair of permutations (sigma,tau) up to simultaneous conjugacy on a pointed set of size 6*n with sigma^3=tau^2=1, acting transitively and with no fixed point. [Vidal]
Also, the asymptotic expansion of the Airy function Ai'(x)/Ai(x) = -sqrt(x) - 1/(4x) + sum_{n>=2} (-1)^n a(n) (4x)^ (1/2-3n/2). [Praehofer]
Maple 6 gives the wrong asymptotics of Ai'(x)=AiryAi(1,x) as x->infty apart from the 3rd term. Therefore asympt(AiryAi(1,x/4)/AiryAi(x/4),x); reproduces only the value a(1)=1 correctly.
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REFERENCES
| S. Janson, The Wiener index of simply generated random trees, Random Structures Algorithms 22 (2003) 337-358.
Michael J. Kearney, Satya N. Majumdar and Richard J. Martin, The first-passage area for drifted Brownian motion and the moments of the Airy distribution, arXiv:0706.2038. [a(n) = 8^n * K_n from Eq. (3)]
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 292.
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LINKS
| S. R. Finch, Shapes of binary trees
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FORMULA
| With offset 1, then a(1) = 1 and, for n > 1, a(n) = (6*n-8)*a(n-1)+sum(a(k)*a(n-k), k=1..n-1) [Praehofer][Martin and Kearney]
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EXAMPLE
| 1 + 5*x + 60*x^2 + 1105*x^3 + 27120*x^4 + 828250*x^5 + 30220800*x^6 + ...
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PROG
| (PARI) {a(n) = local(A); n++; if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6*k - 8) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */
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CROSSREFS
| Pointed version of A012114. Connected pointed version of A012115.
Cf. A060506, A060507, A094199, A121350, A121352, A005133.
Sequence in context: A156125 A128574 A120976 * A113665 A147585 A138215
Adjacent sequences: A062977 A062978 A062979 * A062981 A062982 A062983
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KEYWORD
| nonn,nice,easy
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AUTHOR
| Michael Praehofer (praehofer(AT)ma.tum.de), Jul 24 2001
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EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com) based on comments from Samuel Alexandre Vidal (samuel.vidal(AT)free.fr), Mar 30 2007.
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