A074357
Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,3).
Original entry on oeis.org
0, 0, 0, 0, 0, 30, 168, 639, 2415, 7872, 25542, 77727, 233547, 679410, 1949862, 5490132, 15276456, 41963844, 114153990, 307595853, 822263313, 2181777252, 5751280350, 15069310365, 39269077809, 101817186264, 262776963360
Offset: 0
Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=4, nu(3)=7+3q, nu(4)=19+15q+12q^2, nu(5)=40+45q+42q^2+30q^3+9q^4, so the coefficients of q^3 are 0,0,0,0,0,30.
- M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
- Index entries for linear recurrences with constant coefficients, signature (4,6,-32,-19,96,54,-108,-81).
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nu := proc(b,lambda,n) global q; local qp,i ; if n = 0 then RETURN(1) ; elif n =1 then RETURN(b) ; fi ; qp:=0 ; for i from 0 to n-2 do qp := qp + q^i ; od ; RETURN( b*nu(b,lambda,n-1)+lambda*qp*nu(b,lambda,n-2)) ; end: A074357 := proc(n) RETURN( coeftayl(nu(1,3,n),q=0,3) ) ; end: for n from 0 to 30 do printf("%d,", A074357(n)) ; od ; # R. J. Mathar, Sep 20 2006
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Join[{0, 0, 0}, LinearRecurrence[{4, 6, -32, -19, 96, 54, -108, -81}, {0, 0, 30, 168, 639, 2415, 7872, 25542}, 24]] (* Jean-François Alcover, Sep 22 2017 *)
A074354
Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,2).
Original entry on oeis.org
0, 0, 0, 0, 0, 14, 64, 218, 692, 1982, 5496, 14562, 37692, 95142, 236032, 576074, 1387780, 3304078, 7787656, 18190386, 42151116, 96972534, 221651472, 503650970, 1138286740, 2559944414, 5731095704, 12776843138, 28374100572
Offset: 0
Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=3, nu(3)=5+2q, nu(4)=11+8q+6q^2, nu(5)=21+22q+20q^2+14q^3+4q^4, so the coefficients of q^1 are 0,0,0,0,0,14.
A074353
Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,2).
Original entry on oeis.org
0, 0, 0, 0, 6, 20, 70, 196, 542, 1396, 3526, 8628, 20766, 49092, 114598, 264356, 603998, 1368148, 3076166, 6870740, 15256158, 33696804, 74073510, 162127940, 353460766, 767816500, 1662394310, 3588252916, 7723318942, 16580031876, 35506388646, 75864499428
Offset: 0
Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=3, nu(3)=5+2q, nu(4)=11+8q+6q^2, nu(5)=21+22q+20q^2+14q^3+4q^4, so the coefficients of q^2 are 0,0,0,0,6,20.
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.
- Index entries for linear recurrences with constant coefficients, signature (3,3,-11,-6,12,8).
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LinearRecurrence[{3, 3, -11, -6, 12, 8}, {0, 0, 0, 0, 6, 20, 70, 196}, 50] (* Paolo Xausa, Jan 28 2025 *)
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concat(vector(4), Vec(2*x^4*(3 + x - 4*x^2 - 4*x^3) / ((1 + x)^3*(1 - 2*x)^3) + O(x^40))) \\ Colin Barker, Nov 18 2017
A074356
Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=b*nu(n-1)+lambda*(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(1,3).
Original entry on oeis.org
0, 0, 0, 0, 12, 42, 180, 561, 1833, 5373, 15798, 44367, 123561, 336243, 906054, 2408094, 6344832, 16561824, 42922602, 110472933, 282678423, 719404803, 1822117962, 4594816221, 11540742615, 28880919975, 72033463644, 179107709004
Offset: 0
Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=4, nu(3)=7+3q, nu(4)=19+15q+12q^2, nu(5)=40+45q+42q^2+30q^3+9q^4, so the coefficients of q^2 are 0,0,0,0,12,42.
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nu := proc(n,b,lambda) if n = 0 then 1 ; elif n = 1 then b ; else b*nu(n-1,b,lambda)+lambda*nu(n-2,b,lambda)*add(q^i,i=0..n-2) ; fi ; end: A074356 := proc(n) local b,lambda,thisnu ; b := 1 ; lambda := 3 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,2) ) ; end: for n from 0 to 40 do printf("%d, ",A074356(n) ) ; od ; # R. J. Mathar, Mar 20 2007
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nu[n_, b_, lambda_] := nu[n, b, lambda] = Which[n == 0, 1, n == 1, b, True, b*nu[n - 1, b, lambda] + lambda*nu[n - 2, b, lambda]*Sum[q^i, {i, 0, n - 2}]];
a[n_] := a[n] = Coefficient[nu[n, 1, 3], q, 2];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 30}] (* Jean-François Alcover, Nov 23 2017, from Maple *)
Showing 1-4 of 4 results.
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