A074087 -id:A074087 - cp's OEIS frontend

A074089 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)=(2,3).

0, 0, 0, 0, 0, 78, 501, 2574, 11757, 50034, 203229, 797316, 3046362, 11394774, 41885913, 151732722, 542840175, 1921208586, 6735519249, 23417342568, 80810560596, 277008392478, 943826398893, 3198199361910, 10783017814065

Offset: 0

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

The coefficient of q^0 is A014983(n+1).

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=7, nu(3) = 20 + 6q, nu(4) = 61 + 33q + 21q^2, nu(5) = 182 + 144q + 120q^2 + 78q^3 + 18q^4, so the coefficients of q^3 are 0,0,0,0,0,78.

G. C. Greubel, 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 (8,-12,-40,74,120,-108,-216,-81).

Coefficients of q^0, q^1 and q^2 are in A014983, A074087 and A074088. Related sequences with other values of b and lambda are in A074082-A074086.

m:=25; R:=PowerSeriesRing(Integers(), m); [0,0,0,0,0] cat Coefficients(R!((78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4)); // G. C. Greubel, May 26 2018

b=2; lambda=3; 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: *)
CoefficientList[Series[(78*x^5-123*x^6-498*x^7+297*x^8+1134*x^9 + 567*x^10)/( 1-2*x-3*x^2)^4, {x, 0, 50}], x] (* G. C. Greubel, May 26 2018 *)

x='x+O('x^30); concat([0,0,0,0,0], Vec((78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4)) \\ G. C. Greubel, May 26 2018

G.f.: (78*x^5 -123*x^6 -498*x^7 +297*x^8 +1134*x^9 +567*x^10)/(1 -2*x -3*x^2)^4.

a(n) = 8*a(n-1) -12*a(n-2) -40*a(n-3) +74*a(n-4) +120*a(n-5) -108*a(n-6) -216*a(n-7) -81*a(n-8) for n>=11.

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

A074088 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)=(2,3).

Original entry on oeis.org

0, 0, 0, 0, 21, 120, 585, 2508, 10122, 39042, 145974, 532704, 1907451, 6725004, 23407287, 80591148, 274899288, 930128646, 3124838844, 10432356000, 34634029713, 114403303008, 376184538165, 1231890463020, 4018920819606

Offset: 0

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

The coefficient of q^0 is A014983(n+1).

			The first 6 nu polynomials are nu(0)=1, nu(1)=2, nu(2)=7, nu(3)=20+6q, nu(4)=61+33q+21q^2, nu(5)=182+144q+120q^2+78q^3+18q^4, so the coefficients of q^2 are 0,0,0,0,21,120.

G. C. Greubel, 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 (6,-3,-28,9,54,27).

Coefficients of q^0, q^1 and q^3 are in A014983, A074087 and A074089. Related sequences with other values of b and lambda are in A074082-A074086.

I:=[0,0,21,120,585,2508]; [0,0] cat [n le 6 select I[n] else 6*Self(n-1) -3*Self(n-2) -28*Self(n-3) +9*Self(n-4) +54*Self(n-5) +27*Self(n-6): n in [1..30]]; // G. C. Greubel, May 26 2018

b=2; lambda=3; 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,-3,-28,9,54,27},{0,0,21,120,585,2508},40]] (* Harvey P. Dale, Apr 28 2012 *)

x='x+O('x^30); concat([0,0,0,0], Vec((21*x^4 -6*x^5 -72*x^6 -54*x^7)/(1-2*x-3*x^2)^3)) \\ G. C. Greubel, May 26 2018

G.f.: (21*x^4 -6*x^5 -72*x^6 -54*x^7)/(1-2*x-3*x^2)^3.

a(n) = 6*a(n-1) -3*a(n-2) -28*a(n-3) +9*a(n-4) +54*a(n-5) +27*a(n-6) for n>=8.

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

A074089 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)=(2,3).

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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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A074088 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)=(2,3).

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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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A074089 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)=(2,3).

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).

A074088 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)=(2,3).

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).