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.

A002999 Expansion of (1 + x*exp(x))^2.

Original entry on oeis.org

1, 2, 6, 18, 56, 170, 492, 1358, 3600, 9234, 23060, 56342, 135192, 319514, 745500, 1720350, 3932192, 8912930, 20054052, 44826662, 99614760, 220201002, 484442156, 1061158958, 2315255856, 5033164850, 10905190452, 23555211318, 50734301240, 108984795194, 233538846780
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is the number of binary words of length n where exactly one of each kind of letter that appears is marked. - John Tyler Rascoe, Jul 16 2025

Examples

			a(2) = 6 counts: (1#,1), (1,1#), (1#,2#), (2#,1#), (2#,2), (2,2#) where # denotes a mark. - _John Tyler Rascoe_, Jul 16 2025

Links

Crossrefs

Cf. A048482, A001787, A005183.
Cf. A003013, A001815.

Programs

  • Mathematica
    CoefficientList[Series[1+(2x(7x^3-10x^2+5x-1))/((x-1)^2 (2x-1)^3), {x,0,30}],x]  (* Harvey P. Dale, Apr 04 2011 *)
Table[If[n == 0, 1, (n^2 - n) 2^n/4 + 2*n], {n, 0, 30}] (* T. D. Noe, Apr 04 2011 *)
  • PARI
    A_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace((1+x*exp(x))^2))} \\ John Tyler Rascoe, Jul 16 2025

Formula

From Ralf Stephan, Sep 02 2003: (Start)
a(0) = 1, a(n) = (n^2 - n)*2^n/4 + 2*n.
a(n) = A003013(n) + n = A001815(n) + 2*n. (End)
G.f.: 1+(2*x*(7*x^3-10*x^2+5*x-1))/((x-1)^2*(2*x-1)^3). - Harvey P. Dale, Apr 04 2011

A003013 E.g.f. 1 + x*exp(x) + x^2*exp(2*x).

Original entry on oeis.org

1, 1, 4, 15, 52, 165, 486, 1351, 3592, 9225, 23050, 56331, 135180, 319501, 745486, 1720335, 3932176, 8912913, 20054034, 44826643, 99614740, 220200981, 484442134, 1061158935, 2315255832, 5033164825
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A048482, A001787, A005183.
Cf. A002999, A001815.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1+x Exp[x]+x^2 Exp[2x],{x,0,nn}],x] Range[0,nn]!] (* or *) Join[{1},LinearRecurrence[{8,-25,38,-28,8},{1,4,15,52,165},30]] (* Harvey P. Dale, Nov 01 2011 *)
  • PARI
    a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 8,-28,38,-25,8]^n*[1;1;4;15;52])[1,1] \\ Charles R Greathouse IV, Jun 23 2020

Formula

From Ralf Stephan, Sep 02 2003: (Start)
a(0) = 1, a(n) = (n^2 - n)*2^n/4 + n.
a(n) = A002999(n) - n = A001815(n) + n. (End)
O.g.f.: 1+x*(-1+4*x-8*x^2+6*x^3) / ( (x-1)^2*(2*x-1)^3 ). - R. J. Mathar, Mar 22 2011
a(n) = 8*a(n-1) - 25*a(n-2) + 38*a(n-3) - 28*a(n-4) + 8*a(n-5); a(0)=1, a(1)=1, a(2)=4, a(3)=15, a(4)=52, a(5)=165. - Harvey P. Dale, Nov 01 2011
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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