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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

• T. D. Noe, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (8,-25,38,-28,8)

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