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Search: id:A108242
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| A108242 |
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a(n) is the number of coverings of 1...n by cyclic words of length 3, such that each value from 1 to n appears precisely 3 times. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,1,2,2,2,...,n,n,n}. Repeats of words are allowed in a given covering. |
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4
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| 1, 1, 2, 16, 256, 7184, 311944, 19191448, 1584972224, 169021538944, 22595033625856, 3699135711988736, 727774085471066752, 169399730544125355136, 46039989792346454771456
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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The asymptotic growth of the coefficients is a(n) ~ C (3/2)^n (n!)^2 /n with C approx 0.277
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FORMULA
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Exponential generating function satisfies the linear differential equation: {(6 + 499*t^6 + 270*t^4 + 408*t^8 - 162*t^11 - 558*t^9 - 12*t - 96*t^3 + 66*t^2 - 654*t^7 + 60*t^12 + 154*t^10 - 342*t^5 + 9*t^14)*F(t) + (81*t^10 + 72*t^4 + 198*t^6 + 216*t^8 + 9*t^2)*(diff(diff(F(t), t), t)) + ( - 474*t^6 - 252*t^10 - 6 + 126*t^3 + 594*t^7 - 66*t^2 + 324*t^9 - 54*t^12 - 420*t^8 + 18*t - 264*t^4 + 378*t^5)*(diff(F(t), t)), F(0) = 1}
The a(n) satisfy the recurrence: {a(0) = 1, a(1) = 1, ( - 20779902*n^7 - 134970693*n^6 - 1971620508*n^4 - 2248389*n^8 - 3*n^12 - 4459328640*n - 4242044664*n^3 - 5794678656*n^2 - 618210450*n^5 - 234*n^11 - 1437004800 - 8151*n^10 - 167310*n^9)*a(n) + ( - 7295434560*n - 4550515200 - 914850*n^7 - 5131406304*n^2 - 545289740*n^4 - 2088314700*n^3 - 11400627*n^6 - 95574465*n^5 - 1425*n^9 - 47310*n^8 - 19*n^10)*a(n + 2) + (711103032*n^4 + 8622028800 + 13032306*n^6 + 116250876*n^5 + 2944635984*n^3 + 12385923840*n + 7897844736*n^2 + 18*n^10 + 1404*n^9 + 48708*n^8 + 989496*n^7)*a(n + 3) + ( - 915980400*n - 898128000 - 3060*n^7 - 90090*n^6 - 1499400*n^5 - 15424605*n^4 - 100395540*n^3 - 403611660*n^2 - 45*n^8)*a(n + 4) + (2882376*n^5 + 890994600*n^2 + 2137510944*n + 30916662*n^4 + 210700728*n^3 + 166740*n^6 + 5472*n^7 + 78*n^8 + 2227357440)*a(n + 5) + ( - 1050477120 - 60979*n^6 - 1088733*n^5 - 12105088*n^4 - 27*n^8 - 85853091*n^3 - 379422466*n^2 - 955621272*n - 1944*n^7)*a(n + 6) + (57398400*n + 114*n^6 + 91238400 + 161430*n^4 + 2078100*n^3 + 14985456*n^2 + 6660*n^5)*a(n + 7) + ( - 1225827*n^3 - 58806000 - 63*n^6 - 9078336*n^2 - 92961*n^4 - 3753*n^5 - 35812260*n)*a(n + 8) + (571080*n + 1504800 + 5100*n^3 + 120*n^4 + 81060*n^2)*a(n + 9) + ( - 233178*n - 635976 - 32079*n^2 - 1962*n^3 - 45*n^4)*a(n + 10) + (1116*n + 48*n^2 + 6480)*a(n + 11) + ( - 225*n - 9*n^2 - 1410)*a(n + 12) + 6*a(n + 13),
with a(2) = 2, a(3) = 16, a(4) = 256, a(5) = 7184, a(6) = 311944, a(7) = 19191448, a(8) = 1584972224, a(9) = 169021538944, a(10) = 22595033625856, a(11) = 3699135711988736, a(12) = 727774085471066752}
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EXAMPLE
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a(2)=2 because the two cyclic word coverings are {112, 221} and {111, 222}
a(3)=16: {111 222 333} {111 223 233} {112 122 333} {112 133 223} {113 122 233} {113 123 223} {113 132 223} {112 132 233} {113 133 222} {122 123 133} {122 132 133} {112 123 233} {123 123 123} {123 132 123} {123 132 132} {132 132 132}
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CROSSREFS
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Cf. A052502, A110105, A110106, A110104.
Sequence in context: A009044 A019318 A090727 this_sequence A140307 A114039 A090305
Adjacent sequences: A108239 A108240 A108241 this_sequence A108243 A108244 A108245
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KEYWORD
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nonn
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AUTHOR
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Marni Mishna (marni.mishna(AT)inria.fr), Jun 17 2005
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