A074355 Coefficient of q^1 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).

0, 0, 0, 3, 15, 45, 147, 402, 1134, 2991, 7917, 20367, 52167, 131748, 330876, 824187, 2042763, 5035473, 12361755, 30226614, 73664298, 178971879, 433649769, 1048133619, 2527706127, 6083434824, 14613750648, 35045236083, 83909261319

Offset

0,4

Comments

Coefficient of q^0 is A006130.

Links

Table of n, a(n) for n=0..28.

M. Beattie, S. Dăscălescu and S. Raianu, Lifting of Nichols Algebras of Type B_2, arXiv:math/0204075 [math.QA], 2002.

Formula

G.f.: (9x^4+3x^3)/(1-3x-3x^2)^2 (conjectured). - Ralf Stephan, May 09 2004

Example

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^1 are 0,0,0,3,15,45.

Maple

nu := proc(n, b, lambda) option remember ; 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:
A074355 := proc(n) local b, lambda, thisnu ; b := 1 ; lambda := 3 ; thisnu := nu(n, b, lambda) ; RETURN( coeftayl(thisnu, q=0, 1) ) ; end: # R. J. Mathar, Mar 20 2007

Mathematica

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, 1];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 30}] (* Jean-François Alcover, Nov 23 2017, from 1st Maple program *)

Crossrefs

Coefficient of q^0, q^2 and q^3 are in A006130, A074356 and A074357. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074354, A074358-A074363.

Sequence in context: A048099 A030505 A301632 * A201868 A260021 A005560

Adjacent sequences: A074352 A074353 A074354 * A074356 A074357 A074358

Keyword

nonn

Author

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

Extensions

More terms from R. J. Mathar, Mar 20 2007

Status

approved