A097553
Number of positive words of length n in the monoid Br_6 of positive braids on 7 strands.
Original entry on oeis.org
1, 6, 27, 101, 346, 1131, 3611, 11396, 35761, 111906, 349700, 1092039, 3409031, 10640179, 33206991, 103631414, 323402952, 1009233980, 3149469548, 9828376731, 30670834516, 95712596642, 298684343689, 932085486213, 2908700435744
Offset: 0
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R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1+x^2)^4/(1-6*x+13*x^2-17*x^3+17*x^4-11*x^5+5*x^6-x^7) )); // G. C. Greubel, Apr 20 2021
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CoefficientList[Series[(1+n^2)^4/(1-6n+13n^2-17n^3+17n^4-11n^5+5n^6-n^7),{n,0,30}],n] (* Harvey P. Dale, Sep 27 2019 *)
LinearRecurrence[{6,-13,17,-17,11,-5,1}, {1,6,27,101,346,1131,3611,11396,35761}, 40] (* G. C. Greubel, Apr 20 2021 *)
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def A097553_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (1+x^2)^4/(1-6*x+13*x^2-17*x^3+17*x^4-11*x^5+5*x^6-x^7) ).list()
A097553_list(50) # G. C. Greubel, Apr 20 2021
A097554
Number of positive words of length n in the monoid Br_7 of positive braids on 8 strands.
Original entry on oeis.org
1, 7, 36, 151, 570, 2019, 6893, 23034, 76020, 249077, 812614, 2644447, 8592693, 27895296, 90510106, 293576779, 952053411, 3087093728, 10009389358, 32452403488, 105214363653, 341111617862, 1105895184121, 3585328906357, 11623651559099
Offset: 0
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R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x^2)^5/(1-7*x+18*x^2-25*x^3+24*x^4-15*x^5+6*x^6-x^7) )); // G. C. Greubel, Apr 20 2021
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LinearRecurrence[{7,-18,25,-24,15,-6,1}, {1,7,36,151,570,2019,6893,23034,76020, 249077,812614}, 41] (* G. C. Greubel, Apr 20 2021 *)
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def A097554_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (1+x^2)^5/(1-7*x+18*x^2-25*x^3+24*x^4-15*x^5+6*x^6-x^7) ).list()
A097554_list(40) # G. C. Greubel, Apr 20 2021
A097555
Number of positive words of length n in the monoid Br_8 of positive braids on 9 strands.
Original entry on oeis.org
1, 8, 45, 205, 831, 3133, 11294, 39585, 136302, 464026, 1568151, 5273999, 17681042, 59149925, 197598856, 659479754, 2199585548, 7333198205, 24441067317, 81444567492, 271360676916, 904051477063, 3011711782025, 10032660556567, 33420042561972
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (8,-25,45,-59,57,-41,21,-7,1).
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R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x^2)^6 /(1-8*x+25*x^2-45*x^3+59*x^4-57*x^5+41*x^6-21*x^7+7*x^8-x^9) )); // G. C. Greubel, Apr 20 2021
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LinearRecurrence[{8,-25,45,-59,57,-41,21,-7,1}, {1,8,45,205,831,3133,11294,39585, 136302, 464026, 1568151, 5273999, 17681042}, 41] (* G. C. Greubel, Apr 20 2021 *)
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def A097555_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (1+x^2)^6 /(1-8*x+25*x^2-45*x^3+59*x^4-57*x^5+41*x^6-21*x^7+7*x^8-x^9) ).list()
A097555_list(40) # G. C. Greubel, Apr 20 2021
A097556
Number of positive words of length n in the monoid Br_9 of positive braids on 10 strands.
Original entry on oeis.org
1, 9, 56, 279, 1223, 4932, 18833, 69345, 249166, 880525, 3076295, 10662459, 36749785, 126161246, 431880044, 1475412473, 5032964258, 17150277106, 58395929325, 198723871661, 675989712225, 2298799014859, 7815699898677, 26568450635871
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (9,-32,63,-84,81,-56,27,-8,1).
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R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9) )); // G. C. Greubel, Apr 20 2021
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CoefficientList[Series[(1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9), {x,0,50}], x] (* G. C. Greubel, Apr 20 2021 *)
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def A097556_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9) ).list()
A097556_list(50) # G. C. Greubel, Apr 20 2021