A005744 Expansion of x*(1+x-x^2)/((1-x)^4*(1+x)). (Formerly M3360)

0, 1, 4, 9, 17, 28, 43, 62, 86, 115, 150, 191, 239, 294, 357, 428, 508, 597, 696, 805, 925, 1056, 1199, 1354, 1522, 1703, 1898, 2107, 2331, 2570, 2825, 3096, 3384, 3689, 4012, 4353, 4713, 5092, 5491, 5910, 6350, 6811, 7294, 7799, 8327, 8878, 9453, 10052

Offset

0,3

Comments

Number of n-covers of a 2-set.

Boolean switching functions a(n,s) for s = 2.

Without the initial 0, this is row 1 of the convolution array A213778. - Clark Kimberling, Jun 21 2012

a(n) equals the second column of the triangle A355754. - Eric W. Weisstein, Mar 12 2024

References

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Milan Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8.

A. V. Jayanthan, S. A. Seyed Fakhari, I. Swanson, and S. Yassemi, Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphs, arXiv:2405.06781 [math.AC], 2024. See p. 17.

Vladeta Jovovic, Binary matrices up to row and column permutations.

Index entries for sequences related to Boolean functions

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

a(n) = A002623(n) - (n+1).

a(n) = n*(n-1)/2 + Sum_{j=1..floor((n+1)/2)} (n-2*j+1)*(n-2*j)/2. - N. J. A. Sloane, Nov 28 2003

From R. J. Mathar, Apr 01 2010: (Start)

a(n) = 5*n/12 - 1/16 + 5*n^2/8 + n^3/12 + (-1)^n/16.

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). (End)

a(n) = A181971(n+1, n-1) for n > 0. - Reinhard Zumkeller, Jul 09 2012

a(n) + a(n+1) = A008778(n). - R. J. Mathar, Mar 13 2021

E.g.f.: (x*(2*x^2 + 21*x + 27)*cosh(x) + (2*x^3 + 21*x^2 + 27*x - 3)*sinh(x))/24. - Stefano Spezia, Jul 27 2022

Mathematica

CoefficientList[Series[x (1+x-x^2)/((1-x)^4(1+x)), {x, 0, 50}], x] (* or *) LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 4, 9, 17}, 50] (* Harvey P. Dale, Apr 10 2012 *)

Prog

(PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; -1, 3, -2, -2, 3]^n*[0; 1; 4; 9; 17])[1, 1] \\ Charles R Greathouse IV, Feb 06 2017

Crossrefs

John W. Layman observes that A003453 appears to be the alternating sum transform (PSumSIGN) of A005744.

Cf. A002623, A005745, A005746, A005747, A005748, A005771, A003180, A008778, A052265.

Cf. A181971, A213778.

Cf. A355754.

Sequence in context: A008055 A301019 A137441 * A027367 A348238 A009879

Adjacent sequences: A005741 A005742 A005743 * A005745 A005746 A005747

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Simon Plouffe

Extensions

Additional comments from Alford Arnold

Status

approved