A074083 -id:A074083 - cp's OEIS frontend

A074082 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)=(1,1).

0, 0, 0, 0, 2, 6, 16, 37, 81, 169, 342, 675, 1307, 2491, 4686, 8718, 16066, 29364, 53282, 96065, 172215, 307151, 545286, 963993, 1697701, 2979383, 5211852, 9090060, 15810530, 27429426, 47473828, 81983773, 141286221, 243011173

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

The coefficient of q^0 in nu(n) is the Fibonacci number F(n+1). The coefficient of q^1 is A023610(n-3).

			The first 6 nu polynomials are nu(0)=1, nu(1)=1, nu(2)=2, nu(3)=3+q, nu(4)=5+3q+2q^2, nu(5)=8+7q+6q^2+4q^3+q^4, so the coefficients of q^2 are 0,0,0,0,2,6.

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 (3,0,-5,0,3,1).

Coefficients of q^0, q^1 and q^3 are in A000045, A023610 and A074083. Related sequences with different values of b and lambda are in A074084-A074089.

b=1; lambda=1; expon=2; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0, 0}, LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 6, 16, 37}, 32]] (* Jean-François Alcover, Sep 23 2017 *)

G.f.: (2*x^4-2*x^6-x^7)/(1-x-x^2)^3.

a(n) = 3*a(n-1)-5*a(n-3)+3*a(n-5)+a(n-6) for n>=8.

Edited by Dean Hickerson, Aug 21 2002

A074084 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)=(2,1).

Original entry on oeis.org

0, 0, 0, 2, 9, 32, 102, 306, 883, 2480, 6828, 18514, 49597, 131568, 346194, 904738, 2350695, 6076960, 15641304, 40103778, 102473969, 261046144, 663180222, 1680628946, 4249496795, 10722962256, 27007159428, 67904097074

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

nonn

The coefficient of q^0 is the Pell number A000129(n+1).

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=5, nu(3)=12+2q, nu(4)=29+9q+5q^2, nu(5)=70+32q+24q^2+14q^3+2q^4, so the coefficients of q^1 are 0,0,0,2,9,32.

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, -2, -4, -1).

Coefficients of q^0, q^2 and q^3 are in A000129, A074085 and A074086. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

b=2; lambda=1; expon=1; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0},LinearRecurrence[{4,-2,-4,-1},{0,0,2,9},30]] (* Harvey P. Dale, Apr 18 2012 *)

G.f.: (2x^3+x^4)/(1-2x-x^2)^2.

a(n) = 4a(n-1)-2a(n-2)-4a(n-3)-a(n-4) for n>=5.

Edited by Dean Hickerson, Aug 21 2002

A074086 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)=(2,1).

Original entry on oeis.org

0, 0, 0, 0, 0, 14, 71, 282, 997, 3298, 10439, 32012, 95834, 281494, 814131, 2324422, 6564135, 18362810, 50947395, 140329400, 384031508, 1044880222, 2828084399, 7618214354, 20432838121, 54585196818, 145287466799, 385397215108

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

nonn

The coefficient of q^0 is the Pell number A000129(n+1).

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=5, nu(3)=12+2q, nu(4)=29+9q+5q^2, nu(5)=70+32q+24q^2+14q^3+2q^4, so the coefficients of q^3 are 0,0,0,0,0,14.

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 (8, -20, 8, 26, -8, -20, -8, -1).

Coefficients of q^0, q^1 and q^2 are in A000129, A074084 and A074085. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

b=2; lambda=1; expon=3; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0, 0, 0}, LinearRecurrence[{8, -20, 8, 26, -8, -20, -8, -1}, {0, 0, 14, 71, 282, 997, 3298, 10439}, 25]] (* Jean-François Alcover, Jan 27 2019 *)

G.f.: (14x^5-41x^6-6x^7+49x^8+30x^9+5x^10)/(1-2x-x^2)^4.

a(n) = 8a(n-1)-20a(n-2)+8a(n-3)+26a(n-4)-8a(n-5)-20a(n-6)-8a(n-7)-a(n-8) for n>=11.

Edited by Dean Hickerson, Aug 21 2002

A074085 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) = (2,1).

Original entry on oeis.org

0, 0, 0, 0, 5, 24, 91, 308, 978, 2978, 8802, 25440, 72251, 202316, 559941, 1534548, 4170256, 11250630, 30158900, 80389600, 213204513, 562896832, 1480086111, 3877337556, 10123000126, 26347306474, 68378847990, 176994780672

Offset: 0

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 19 2002

nonn

The coefficient of q^0 is the Pell number A000129(n+1).

			The first 6 nu polynomials are nu(0) = 1, nu(1) = 2, nu(2) = 5, nu(3) = 12 + 2*q, nu(4) = 29 + 9*q + 5*q^2, nu(5) = 70 + 32q + 24*q^2 + 14*q^3 + 2*q^4, so the coefficients of q^2 are 0,0,0,0,5,24.

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, -9, -4, 9, 6, 1).

Coefficients of q^0, q^1 and q^3 are in A000129, A074084 and A074086. Related sequences with other values of b and lambda are in A074082-A074083 and A074087-A074089.

b=2; lambda=1; expon=2; nu[0]=1; nu[1]=b; nu[n_] := nu[n]=Together[b*nu[n-1]+lambda(1-q^(n-1))/(1-q)nu[n-2]]; a[n_] := Coefficient[nu[n], q, expon]
(* Second program: *)
Join[{0,0}, LinearRecurrence[{6, -9, -4, 9, 6, 1}, {0, 0, 5, 24, 91, 308}, 30]] (* Jean-François Alcover, Dec 13 2018 *)

G.f.: (5*x^4 - 6*x^5 - 8*x^6 - 2*x^7)/(1 - 2*x - x^2)^3.

a(n) = 6*a(n-1) - 9*a(n-2) - 4*a(n-3) + 9*a(n-4) + 6*a(n-5) + a(n-6) for n >= 8.

Edited by Dean Hickerson, Aug 21 2002

cp's OEIS Frontend

A074082 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)=(1,1).

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A074084 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)=(2,1).

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A074086 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)=(2,1).

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A074085 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) = (2,1).

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Mathematica

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Extensions

A074082 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)=(1,1).

A074084 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)=(2,1).

A074086 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)=(2,1).

A074085 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) = (2,1).