A074361 -id:A074361 - cp's OEIS frontend

A074363 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(3,1).

0, 0, 0, 0, 0, 36, 246, 1293, 6057, 26592, 111934, 457353, 1827529, 7176636, 27789976, 106371588, 403204880, 1515647250, 5656172420, 20974163475, 77339044883, 283743384228, 1036296662574, 3769287797151, 13658724680991, 49325767966842, 177570110818794

Offset: 0

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

Coefficient of q^0 is A006190(n+1).

			The first 6 nu polynomials are nu(0)=1, nu(1)=3, nu(2)=10, nu(3)=33+3q, nu(4)=109+19q+10q^2, nu(5)=360+93q+66q^2+36q^3+3q^4, so the coefficients of q^1 are 0,0,0,0,0,36.

Colin Barker, Table of n, a(n) for n = 0..1001
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 (12,-50,72,21,-72,-50,-12,-1).

Coefficients of q^0, q^1 and q^2 are in A006190, A074361 and A074362. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074360.

concat(vector(4), Vec(x^4*(3 + x)*(12 - 66*x + 69*x^2 + 60*x^3 + 10*x^4) / (1 - 3*x - x^2)^4 + O(x^40))) \\ Colin Barker, Nov 18 2017

From Colin Barker, Nov 18 2017: (Start)
G.f.: x^4*(3 + x)*(12 - 66*x + 69*x^2 + 60*x^3 + 10*x^4) / (1 - 3*x - x^2)^4.
a(n) = 12*a(n-1) - 50*a(n-2) + 72*a(n-3) + 21*a(n-4) - 72*a(n-5) - 50*a(n-6) - 12*a(n-7) - a(n-8) for n>9.
(End)

More terms from Brent Lehman (mailbjl(AT)yahoo.com), Aug 25 2002
Missing a(0)=0 inserted by Sean A. Irvine, Jan 20 2025
Missing a(0)=0 inserted in b-file by David Radcliffe, Aug 01 2025

A074358 Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n >= 2, nu(n) = bnu(n-1) + lambda(1 + q + q^2 + ... + q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 4, 20, 80, 288, 976, 3184, 10112, 31488, 96576, 292672, 878336, 2614784, 7731456, 22728448, 66482176, 193617920, 561718272, 1624101888, 4681535488, 13457924096, 38592008192, 110419341312, 315287830528, 898583560192

Offset: 0

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

nonn

Coefficient of q^0 is A002605.

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3) = 16 + 4q, nu(4) = 44 + 20q + 12q^2, nu(5) = 120 + 80q + 64q^2 + 40q^3 + 8q^4, so the coefficients of q^1 are 0,0,0,4,20,80.

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

Coefficient of q^0, q^2 and q^3 are in A002605, A074359 and A074360. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

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: A074358 := proc(n) RETURN( coeftayl(nu(2,2,n),q=0,1) ) ; end: for n from 0 to 30 do printf("%d,", A074358(n)) ; od ; # R. J. Mathar, Sep 20 2006

nu[0] = 1; nu[1] = 2; nu[n_] := nu[n] = 2*nu[n-1] + 2*Total[q^Range[0, n-2] ]*nu[n-2] // Expand;
a[n_] := Coefficient[nu[n], q, 1];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Nov 17 2017 *)

G.f.: 4*x^3*(x + 1)/(2*x^2 + 2*x - 1)^2 (conjectured). - Chai Wah Wu, May 30 2016
a(n) = (1/18)*((1 + sqrt(3))^n*(-9 + 2*sqrt(3)) - (1 - sqrt(3))^n*(9 + 2*sqrt(3)) + 3*((1 - sqrt(3))^n + (1 + sqrt(3))^n)*n) for n > 0 (conjectured). - Colin Barker, Nov 17 2017
a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n > 4 (conjectured). - Colin Barker, Nov 17 2017

More terms from R. J. Mathar, Sep 20 2006

A074360 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 0, 0, 40, 232, 1072, 4400, 16864, 61728, 218496, 753792, 2547840, 8468608, 27755776, 89886976, 288101888, 915089920, 2883416064, 9021001728, 28042881024, 86672025600, 266472878080, 815347462144, 2483820617728

Offset: 0

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

nonn

Coefficient of q^0 is A002605.

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3)=16+4q, nu(4)=44+20q+12q^2, nu(5)=120+80q+64q^2+40q^3+8q^4, so the coefficients of q^1 are 0,0,0,0,0,40.

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

Coefficient of q^0, q^1 and q^2 are in A002605, A074358 and A074359. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

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: A074360 := proc(n) RETURN( coeftayl(nu(2,2,n),q=0,3) ) ; end: for n from 0 to 30 do printf("%d,", A074360(n)) ; od ; # R. J. Mathar, Sep 20 2006

nu[0] = 1; nu[1] = 2; nu[n_] := nu[n] = 2*nu[n - 1] + 2*Total[q^Range[0, n - 2]]*nu[n - 2] // Expand;
a[n_] := Coefficient[nu[n], q, 3];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 18 2017 *)

Conjecture: O.g.f: 8*x^5*(1+x)*(12*x^4+24*x^3-2*x^2-16*x+5)/(2*x^2+2*x-1)^4. - R. J. Mathar, Jul 22 2009

More terms from R. J. Mathar, Sep 20 2006

A074359 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

Original entry on oeis.org

0, 0, 0, 0, 12, 64, 280, 1088, 3968, 13856, 46912, 155136, 503616, 1610496, 5086336, 15895552, 49229312, 151275008, 461662208, 1400356864, 4224703488, 12683452416, 37911164928, 112865394688, 334788444160, 989756825600

Offset: 0

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

nonn

Coefficient of q^0 is A002605.

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=6, nu(3)=16+4q, nu(4)=44+20q+12q^2, nu(5)=120+80q+64q^2+40q^3+8q^4, so the coefficients of q^2 are 0,0,0,0,12,64.

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 (6, -6, -16, 12, 24, 8).

Coefficient of q^0, q^1 and q^3 are in A002605, A074358 and A074360. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074357, A074361-A074363.

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: A074359 := proc(n) local b,lambda,thisnu ; b := 2 ; lambda := 2 ; thisnu := nu(n,b,lambda) ; RETURN( coeftayl(thisnu,q=0,2) ) ; end: for n from 0 to 40 do printf("%d, ",A074359(n) ) ; od ; # R. J. Mathar, Mar 20 2007

Join[{0, 0}, LinearRecurrence[{6, -6, -16, 12, 24, 8}, {0, 0, 12, 64, 280, 1088}, 24]] (* Jean-François Alcover, Sep 23 2017 *)

Conjecture: O.g.f: 4*x^4*(-3+2*x+8*x^2+4*x^3)/(2*x^2+2*x-1)^3. - R. J. Mathar, Jul 22 2009

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

A074362 Coefficient of q^2 in nu(n), where nu(0) = 1, nu(1) = b and, for n >= 2, nu(n) = bnu(n-1) + lambda(1 + q + q^2 + ... + q^(n-2))*nu(n-2) with (b,lambda) = (3,1).

Original entry on oeis.org

0, 0, 0, 0, 10, 66, 336, 1527, 6513, 26667, 106102, 413265, 1583331, 5986689, 22392606, 83002842, 305308666, 1115587020, 4052786850, 14648359515, 52705460583, 188868467853, 674332868566, 2399653030899, 8513523719661

Offset: 0

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

nonn

Coefficient of q^0 is A006190(n+1).

			The first 6 nu polynomials are nu(0) = 1, nu(1) = 3, nu(2) = 10, nu(3) = 33 + 3*q, nu(4) = 109 + 19*q + 10*q^2, nu(5) = 360 + 93*q + 66*q^2 + 36*q^3 + 3*q^4, so the coefficients of q^1 are 0,0,0,0,10,66.

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 (9, -24, 9, 24, 9, 1).

Coefficient of q^0, q^1 and q^3 are in A006190, A074361 and A074363. Related sequences with other values of b and lambda are in A074082-A074089, A074352-A074360.

Join[{0, 0}, LinearRecurrence[{9, -24, 9, 24, 9, 1}, {0, 0, 10, 66, 336, 1527}, 30]] (* Jean-François Alcover, Dec 13 2018 *)

G.f.: (-3*x^7 - 18*x^6 - 24*x^5 + 10*x^4)/(1 - 3*x - x^2)^3.

More terms from Brent Lehman (mailbjl(AT)yahoo.com), Aug 25 2002

cp's OEIS Frontend

A074363 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)=(3,1).

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A074358 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)=(2,2).

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A074360 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)=(2,2).

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A074359 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)=(2,2).

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Maple

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A074362 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) = (3,1).

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Programs

Mathematica

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Extensions

A074363 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(3,1).

A074358 Coefficient of q^1 in nu(n), where nu(0)=1, nu(1)=b and, for n >= 2, nu(n) = bnu(n-1) + lambda(1 + q + q^2 + ... + q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

A074360 Coefficient of q^3 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

A074359 Coefficient of q^2 in nu(n), where nu(0)=1, nu(1)=b and, for n>=2, nu(n)=bnu(n-1)+lambda(1+q+q^2+...+q^(n-2))*nu(n-2) with (b,lambda)=(2,2).

A074362 Coefficient of q^2 in nu(n), where nu(0) = 1, nu(1) = b and, for n >= 2, nu(n) = bnu(n-1) + lambda(1 + q + q^2 + ... + q^(n-2))*nu(n-2) with (b,lambda) = (3,1).