A074352 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)=(1,2).

0, 0, 0, 2, 8, 22, 60, 146, 352, 814, 1860, 4170, 9256, 20326, 44300, 95874, 206320, 441758, 941780, 2000058, 4233144, 8932310, 18796700, 39457522, 82643328, 172743182, 360399460, 750625066, 1560902472, 3241109574, 6720828460, 13918875490, 28792188176

Offset

0,4

Comments

Coefficient of q^0 is A001045(n+1).

Links

Colin Barker, 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 (2,3,-4,-4).

Formula

a(0) = 0; for n>0, a(n) = (1/27)*(2^n*(6*n-11) + (-1)^n*(6*n-16)).

From Colin Barker, Nov 18 2017: (Start)

G.f.: 2*x^3*(1 + 2*x) / ((1 + x)^2*(1 - 2*x)^2).

a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 4*a(n-4) for n>4. (End)

Example

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 0,0,0,2,8,22.

Mathematica

LinearRecurrence[{2, 3, -4, -4}, {0, 0, 0, 2, 8}, 50] (* Paolo Xausa, Jan 28 2025 *)

Prog

(PARI) a(n)=if(n<1, 0, (1/27)*(2^n*(6*n-11)+(-1)^n*(6*n-16)))
(PARI) a(n)=if(n<1, 0, (1/81)*(2^(n-1)*(6*n^2-43)+ (-1)^n*(6*n^2-24*n+62)))
(PARI) concat(vector(3), Vec(2*x^3*(1 + 2*x) / ((1 + x)^2*(1 - 2*x)^2) + O(x^40))) \\ Colin Barker, Nov 18 2017

Crossrefs

Coefficient of q^0, q^2 and q^3 are in A001045, A074353 and A074354. Related sequences with other values of b and lambda are in A074082-A074089, A074353-A074363.

Sequence in context: A145654 A221880 A295141 * A301555 A261561 A017928

Adjacent sequences: A074349 A074350 A074351 * A074353 A074354 A074355

Keyword

nonn,easy,changed

Author

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

Extensions

More terms and formula from Benoit Cloitre, Jan 12 2003

Corrected by Franklin T. Adams-Watters, Oct 25 2006

Corrected by T. D. Noe, Oct 25 2006

Status

approved