OFFSET
1,2
COMMENTS
A cover of a set S is a collection of nonempty subsets of S whose union is S. A cover of S is called minimal if none of its proper subsets covers S. [from the Hearne/Wagner reference]
Partial sums of A053221.
Construct an inverted triangle table with n rows as follows: the first row are numbers from 1 to n; for the other rows, each number is the sum of the two numbers above it. Then a(n) is the sum of all numbers in the table. See examples below. - Jianing Song, Sep 04 2018
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
Anthony J. Macula, Lewis Carroll and the Enumeration of Minimal Covers, Math. Mag. vol. 68, n4, p 274 Oct '95.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (7,-19,25,-16,4)
FORMULA
G.f.: x*(1 - x - x^2)/((1 - x)^3*(1 - 2*x)^2).
a(n) = (n + 1)*2^n - (n + 1)*(n + 2)/2. - Paul Barry, Jan 27 2003
E.g.f.: (2*x + 1)*exp(2*x) - (x^2/2 + 2*x + 1)*exp(x). - Jianing Song, Sep 04 2018
EXAMPLE
From Jianing Song, Sep 04 2018: (Start)
For n = 4 the inverted triangle table is:
1 2 3 4
3 5 7
8 12
20
So a(4) = 1 + 2 + 3 + 4 + 3 + 5 + 7 + 8 + 12 + 20 = 65.
For n = 5 the inverted triangle table is:
1 2 3 4 5
3 5 7 9
8 12 16
20 28
48
So a(5) = 1 + 2 + 3 + 4 + 5 + 3 + 5 + 7 + 9 + 8 + 12 + 16 + 20 + 28 + 48 = 171. (End)
MAPLE
a := n -> add((n+1)*binomial(n+1, k+1)/2, k=1..n):
seq(a(n), n=1..30); # Zerinvary Lajos, May 08 2007
A003469:=(-1+z+z**2)/(2*z-1)**2/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(n+1)2^n-(n+1)(n+2)/2, {n, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 30 2011 *)
CoefficientList[Series[((2*x + 1)*Exp[2*x] - (x^2/2 + 2*x + 1)*Exp[x])/x, {x, 0, 200}], x]*Table[(k+1)!, {k, 0, 200}] (* Stefano Spezia, Sep 04 2018 *)
PROG
(PARI) a(n) = (n+1)*2^n-(n+1)*(n+2)/2;
(Magma) [2^n*(n+1)-(n^2+3*n+2)/2: n in [1..30]]; // Vincenzo Librandi, Aug 19 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Offset changed from 2 to 1 by Vincenzo Librandi, Aug 19 2011
Title corrected by Geoffrey Critzer, Jun 29 2013
STATUS
approved