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.

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A060932 Fifth convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 18, 159, 942, 4311, 16536, 55898, 171924, 491487, 1325546, 3409347, 8430246, 20164223, 46880424, 106350942, 236147828, 514553154, 1102562952, 2327442276, 4847463408, 9974081130, 20297335340
Offset: 0

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Author

Wolfdieter Lang, Apr 20 2001

Keywords

Crossrefs

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40);
    Coefficients(R!( ((1+2*x)/(1-x-x^2))^6 )); // G. C. Greubel, Apr 08 2021
    
  • Mathematica
    Table[((744+2990*n+2895*n^2+1925*n^3+825*n^4+125*n^5)*LucasL[n+2] +3*(256+390*n + 505*n^2+425*n^3+175*n^4+25*n^5)*LucasL[n+1])/(5^2*5!), {n,0,40}] (* G. C. Greubel, Apr 08 2021 *)
  • Sage
    def A060932_list(prec):
        P. = PowerSeriesRing(ZZ, prec)
        return P( ((1+2*x)/(1-x-x^2))^6 ).list()
    A060932_list(40) # G. C. Greubel, Apr 08 2021

Formula

a(n) = A060922(n+5, 5) (sixth column of Lucas triangle).
G.f.: ((1+2*x)/(1-x-x^2))^6.
a(n) = ( 25*(125*n^5 +825*n^4 +1925*n^3 +2895*n^2 +2990*n +744)*L(n+2) +(1875*n^5 +13125*n^4 +31875*n^3 +37875*n^2 +29250*n +19200)*L(n+1))/(5!*5^4), with the Lucas numbers L(n)=A000032(n).

A060933 Sixth convolution of Lucas numbers A000032(n+1), n >= 0.

Original entry on oeis.org

1, 21, 217, 1498, 7910, 34566, 131446, 449732, 1416513, 4174765, 11651717, 31075422, 79751854, 198036146, 477899790, 1124785648, 2589534248, 5845989156, 12968091584, 28316428700, 60953528230, 129515454530, 271955244610, 564879359940, 1161646929275, 2366938010983, 4781794056543
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Crossrefs

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40);
    Coefficients(R!( ((1+2*x)/(1-x-x^2))^7 )); // G. C. Greubel, Apr 08 2021
    
  • Maple
    m:= 40; S:= series( ((1+2*x)/(1-x-x^2))^7, x, m+1);
    seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Apr 08 2021
  • Mathematica
    Table[(n+1)(2(100n^5+845n^4+2480n^3+4345n^2+5910n+2952)LucasL[n+2]+(125n^5+ 1030n^4+2995n^3+5930n^2+8280n+288)LucasL[n+1])/18000,{n,0,30}] (* Harvey P. Dale, Aug 13 2013 *)
    CoefficientList[Series[((1+2x)/(1-x-x^2))^7, {x,0,30}], x] (* Vincenzo Librandi, Aug 13 2013 *)
  • Sage
    def A060930_list(prec):
        P. = PowerSeriesRing(ZZ, prec)
        return P( ((1+2*x)/(1-x-x^2))^7 ).list()
    A060930_list(40) # G. C. Greubel, Apr 08 2021

Formula

G.f.: ( (1+2*x) / (1-x-x^2) )^7.
a(n) = A060922(n+6, 6) (seventh column of Lucas triangle).
a(n) = (n+1)*(2*(100*n^5 +845*n^4 +2480*n^3 +4345*n^2 +5910*n +2952)*L(n+2) + (125*n^5 +1030*n^4 +2995*n^3 +5930*n^2 +8280*n +288)*L(n+1))/(6!*5^2), with the Lucas numbers L(n)=A000032(n).
Showing 1-2 of 2 results.