A352971 Moments of the distribution of position of the first occurrence of pattern aa in a random ternary word.
1, 12, 258, 8274, 353742, 18904602, 1212354798, 90706565514, 7756033173342, 746093257148442, 79745110236049038, 9375786203927344554, 1202540991574287431742, 167091435183140588426682, 25003060551369349424359278, 4008624526767825553573112394
Offset: 0
Keywords
Links
- Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 59.
Crossrefs
Cf. A302922.
Programs
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Maple
g := exp(2*x)/(9-6*exp(x)-2*exp(2*x)) ; taylor(g,x=0,40) ; L := gfun[seriestolist](%) ; seq( op(i,L)*(i-1)!,i=1..nops(L)) ; # R. J. Mathar, Mar 02 2023
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Mathematica
nn = 15; c[z_] := (1 - z^k)/(1 - z); T[z_] := z^k/(z^k + (1 - m z) c[z]); G[t_] := T[t/m]; Range[0, nn]! CoefficientList[Series[G[Exp[t]] /. {k -> 2, m -> 3}, {t, 0, nn}],t] -
PARI
seq(n)=my(p=exp(x + O(x*x^n))); Vec(serlaplace(p^2/(9 - 6*p - 2*p^2))) \\ Andrew Howroyd, May 06 2023
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SageMath
# uses[egfExpand from A362718] def egf(x): return exp(2*x)/(9 - 6*exp(x) - 2*exp(2*x)) print(egfExpand(egf, 1, 15)) # Peter Luschny, May 06 2023
Formula
E.g.f.: exp(2*t)/(9 - 6*exp(t) - 2*exp(2*t)).
a(n) ~ n! * (3 - sqrt(3)) / (12 * (log(3*(sqrt(3) - 1)/2))^(n+1)). - Vaclav Kotesovec, Apr 13 2022
Extensions
Typo in a(7) corrected by Georg Fischer, May 06 2023
Comments