A002697 a(n) = n*4^(n-1).
0, 1, 8, 48, 256, 1280, 6144, 28672, 131072, 589824, 2621440, 11534336, 50331648, 218103808, 939524096, 4026531840, 17179869184, 73014444032, 309237645312, 1305670057984, 5497558138880, 23089744183296
Offset: 0
Examples
From _Bernard Schott_, Jan 04 2013: (Start)
See the comment about intersection of X and Y.
If A={b,c}, then in P(A) we have:
{b}Inter{b}={b},
{b}Inter{b,c}={b},
{c}Inter{c}={c},
{c}Inter{b,c}={c},
{b,c}Inter{b}={b},
{b,c}Inter{c}={c},
{b,c}Inter{b,c}={b,c}
and : #{b}+ #{b}+ #{c}+ #{c}+ #{b}+ #{c}+ #{b,c} = 8 = 2*4^(2-1) = a(2).
The other intersections are empty.
(End)
References
- Miklos Bona, Combinatorics of Permutations, Chapman and Hall/CRC, 2004, pp. 1, 43, 64.
- C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 516.
- 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
- Danny Rorabaugh, Table of n, a(n) for n = 0..1000
- F. Ellermann, Illustration of binomial transforms
- Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 414
- Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
- Constantinos Kourouzides, A double counting argument on the hypercube graph
- 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)
- Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
- 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
- Eric Weisstein's World of Mathematics, Hypercube Graph
- Eric Weisstein's World of Mathematics, Wiener Index
- Index entries for linear recurrences with constant coefficients, signature (8,-16).
Crossrefs
Programs
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Maple
A002697:=1/(4*z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation A002697:=n->n*4^(n-1): seq(A002697(n), n=0..30); # Wesley Ivan Hurt, Mar 30 2014
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Mathematica
Table[n 4^(n - 1), {n, 0, 30}] (* Harvey P. Dale, Jan 18 2012 *) LinearRecurrence[{8, -16}, {0, 1}, 30] (* Harvey P. Dale, Jan 18 2012 *) CoefficientList[Series[x/(1 - 4 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *) -
PARI
a(n)=if(n<0,0,n*4^(n-1))
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Sage
[n*4^(n-1) for n in range(22)] # Danny Rorabaugh, Mar 27 2015
Formula
a(n) = n*4^(n-1).
G.f.: x/(1-4x)^2. a(n+1) is the convolution of powers of 4 (A000302). - Wolfdieter Lang, May 16 2003
Third binomial transform of n. E.g.f.: x*exp(4x). - Paul Barry, Jul 22 2003
a(n) = Sum_{k=0..n} k*binomial(2*n, 2*k). - Benoit Cloitre, Jul 30 2003
For n>=0, a(n+1) = Sum_{i+j+k+l=n} binomial(2i, i)*binomial(2j, j)*binomial(2k, k)*binomial(2l, l). - Philippe Deléham, Jan 22 2004
a(n) = Sum_{k=0..n} 4^(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2. - Paul Barry, Oct 15 2004
Sum_{n>0} 1/a(n) = 8*log(2) - 4*log(3). - Jaume Oliver Lafont, Sep 11 2009
a(0) = 0, a(n) = 4*a(n-1) + 4^(n-1). - Vincenzo Librandi, Dec 31 2010
a(0) = 0, a(1) = 1, a(n) = 8*a(n-1) - 16*a(n-2). - Harvey P. Dale, Jan 18 2012
G.f.: W(0)*x/2 , where W(k) = 1 + 1/( 1 - 4*x*(k+2)/( 4*x*(k+2) + (k+1)/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(5/4). - Amiram Eldar, Oct 28 2020
a(n) = (1/2)*Sum_{k=0..n} k*binomial(2*n, k). Compare this with the formula of Benoit Cloitre above. - Wolfdieter Lang, Nov 12 2021
a(n) = (-1)^(n-1)*det(M(n)) for n > 0, where M(n) is the n X n symmetric Toeplitz matrix whose first row consists of 1, 3, ..., 2*n-1. - Stefano Spezia, Aug 04 2022
Comments