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

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy
• nice

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 Janjić, 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).

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.

Programs

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 *)

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

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

Extensions

Additional comments from Alford Arnold