cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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).

Original entry on oeis.org

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

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Author

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

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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.

Extensions

Edited by Dean Hickerson, Aug 21 2002

A074087 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)=(2,3).

Original entry on oeis.org

0, 0, 0, 6, 33, 144, 570, 2118, 7587, 26448, 90420, 304470, 1013061, 3338112, 10911150, 35423862, 114342855, 367242336, 1174368360, 3741029094, 11876859369, 37591894320, 118659631650, 373630740966, 1173847761003
Offset: 0

Views

Author

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

Keywords

Comments

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

Examples

			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^1 are 0,0,0,6,33,144.

Links

Crossrefs

Coefficients of q^0, q^2 and q^3 are in A014983, A074088 and A074089. Related sequences with other values of b and lambda are in A074082-A074086.

Programs

  • Magma
    I:=[0,0,6,33]; [0] cat [n le 4 select I[n] else 4*Self(n-1) + 2*Self(n-2) -12*Self(n-3) -9*Self(n-4): n in [1..30]]; // G. C. Greubel, May 26 2018
  • Mathematica
    b=2; lambda=3; 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,-12,-9}, {0,0,6,33}, 50]] (* G. C. Greubel, May 26 2018 *)
  • PARI
    x='x+O('x^30); concat([0,0,0], Vec((6*x^3 +9*x^4)/(1-2*x-3*x^2)^2)) \\ G. C. Greubel, May 26 2018

Formula

G.f.: (6*x^3 +9*x^4)/(1-2*x-3*x^2)^2.
a(n) = 4*a(n-1) +2*a(n-2) -12*a(n-3) -9*a(n-4) for n>=5.

Extensions

Edited by Dean Hickerson, Aug 21 2002
Showing 1-2 of 2 results.

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