A003013 -id:A003013 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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

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)

Cf. A048482, A001787, A005183.

Cf. A003013, A001815.

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 *)

A_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace((1+x*exp(x))^2))} \\ John Tyler Rascoe, Jul 16 2025

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

A048482 a(n) = T(n,n), array T given by A048472.

Original entry on oeis.org

1, 3, 13, 49, 161, 481, 1345, 3585, 9217, 23041, 56321, 135169, 319489, 745473, 1720321, 3932161, 8912897, 20054017, 44826625, 99614721, 220200961, 484442113, 1061158913, 2315255809, 5033164801, 10905190401, 23555211265, 50734301185, 108984795137

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8).

Cf. A002999, A003013, A001787, A005183, A001788.

[(n^2+n)*2^(n-1) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011

LinearRecurrence[{7,-18,20,-8},{1,3,13,49},30] (* Harvey P. Dale, Feb 02 2015 *)

Vec(-(8*x^3-10*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Nov 26 2014

a(n) = 2 * A001788(n) + 1.
a(n) = (n^2+n)*2^(n-1) + 1. - Ralf Stephan, Sep 02 2003
G.f.: -(8*x^3-10*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Nov 26 2014
a(0)=1, a(1)=3, a(2)=13, a(3)=49, a(n)=7*a(n-1)-18*a(n-2)+ 20*a(n-3)- 8*a(n-4). - Harvey P. Dale, Feb 02 2015

cp's OEIS Frontend

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

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