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Search: id:A136439
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| A136439 |
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Sum of heights of all 1-watermelons with wall of length 2*n. |
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+0
1
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| 1, 3, 10, 34, 118, 417, 1495, 5421, 19838, 73149, 271453, 1012872, 3797228, 14294518, 54006728, 204702328, 778115558, 2965409556, 11327549778, 43361526366, 166306579062, 638969153207, 2458973656584, 9477124288144
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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a(n) is the sum of heights of all Dyck excursions of length 2*n (nonnegative walks beginning and ending at 0 with jumps -1,+1).
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REFERENCES
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N. G. de Bruijn, D. E. Knuth and S. O. Rice, The average height of planted plane trees, in: Graph Theory and Computing (ed. T. C. Read), Academic Press, New York, 1972, pp. 15-22.
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LINKS
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M. Fulmek, Asymptotics of the average height of 2-watermelons with a wall, Elec. J. Combin. 14 (2007) R64
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FORMULA
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G.f.=Sum(k*(H[k]-H[k-1]),k=1..infinity), where H[0]=1 and H[k]=1/(1-zH[k-1]) for k=1,2,... (the Maple program makes use of this g.f.). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 13 2008
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MAPLE
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H[0]:=1: for k to 30 do H[k]:=simplify(1/(1-z*H[k-1])) end do: g:=sum(j*(H[j]-H[j-1]), j=1..30): gser:=series(g, z=0, 27): seq(coeff(gser, z, n), n=1..24); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 13 2008
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MATHEMATICA
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c[n_] := (2*n)!/(n!*(n+1)!) s[n_, a_] := Sum[If[k < 1, 0, DivisorSigma[0, k]*Binomial[2*n, n+a-k]/Binomial[2*n, n]], {k, a-n, a+n}] h[n_] := (n+1)*(s[n, 1]-2*s[n, 0]+s[n, -1]) - 1 a[n_] := h[n]*c[n]
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CROSSREFS
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Cf. A000108, A008549, A078920.
Sequence in context: A071725 A026016 A109263 this_sequence A047017 A026616 A047032
Adjacent sequences: A136436 A136437 A136438 this_sequence A136440 A136441 A136442
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KEYWORD
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nonn
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AUTHOR
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S. R. Finch (Steven.Finch(AT)inria.fr), Apr 02 2008
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