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A002699 a(n) = n*2^(2*n-1).
(Formerly M2090 N0825)
8
0, 2, 16, 96, 512, 2560, 12288, 57344, 262144, 1179648, 5242880, 23068672, 100663296, 436207616, 1879048192, 8053063680, 34359738368, 146028888064, 618475290624, 2611340115968, 10995116277760, 46179488366592, 193514046488576 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Right side of binomial sum Sum(i * binomial(2*n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
Coefficients of shifted Chebyshev polynomials.
Starting with offset 1 = 4th binomial transform of [2, 8, 0, 0, 0, ...]. - Gary W. Adamson, Jul 21 2009
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the sum of the size of the symmetric difference of x and y for every (x,y) of B. - Ross La Haye, Jan 04 2013
It's the relation [27] with T(n) in the document of Ross. Following the last comment of Ross, A002697 is the similar sequence when replacing "symmetric difference" by "intersection" and A212698 is the similar sequence when replacing "symmetric difference" by union. - Bernard Schott, Jan 04 2013
If Delta = Symmetric difference, here, X Delta Y and Y Delta X are considered as two distinct Cartesian products, if we want to consider that X Delta Y = X Delta Y is the same Cartesian product, see A002697. - Bernard Schott, Jan 15 2013
REFERENCES
C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Rebecca Bourn and William Q. Erickson, A palindromic polynomial connecting the earth mover's distance to minuscule lattices of Type A, arXiv:2307.02652 [math.CO], 2023.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
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
FORMULA
a(n) = 2 * A002697(n). - Bernard Schott, Jan 04 2013
a(n) = A212698(n) - A002697(n)
a(n) = 8*a(n-1)-16*a(n-2) with n>1, a(0)=0, a(1)=2. - Vincenzo Librandi, Mar 20 2013
G.f.: (2*x)/(1 - 4*x)^2. - Harvey P. Dale, Jul 28 2021
E.g.f.: (exp(4*x) - 1)/2. - Stefano Spezia, Aug 04 2022
MAPLE
A002699 := n->n*2^(2*n-1);
A002699:=2*z/(4*z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(n 2^(2 n - 1)), {n, 0, 30}] (* Vincenzo Librandi, Mar 20 2013 *)
LinearRecurrence[{8, -16}, {0, 2}, 30] (* Harvey P. Dale, Dec 20 2015 *)
PROG
(Magma) [n*2^(2*n-1): n in [0..30]]; /* or */ I:=[0, 2]; [n le 2 select I[n] else 8*Self(n-1)-16*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 20 2013
(PARI) a(n)=n*2^(2*n-1) \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
Found in A053125 and A053124.
Sequence in context: A141243 A163229 A038749 * A335349 A005058 A082639
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)