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A002697
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n*4^(n-1).
(Formerly M4534 N1923)
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30
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0, 1, 8, 48, 256, 1280, 6144, 28672, 131072, 589824, 2621440, 11534336, 50331648, 218103808, 939524096, 4026531840, 17179869184, 73014444032, 309237645312, 1305670057984, 5497558138880, 23089744183296
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Coefficient of x^(2n-2) in Chebyshev polynomial T(2n) is -a(n).
Let M_n be the n X n matrix m_(i,j)=1+2*abs(i-j) then det(M_n)=(-1)^(n-1)*a(n-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 28 2002
Number of subsequences 00 in all words of length n+1 on the alphabet {0,1,2,3}. Example: a(2)=8 because we have 000,001,002,003,100,200,300 (the other 57=A125145(3) words of length 3 have no subsequences 00). a(n)=Sum(k*A128235(n+1,k),k=0..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 27 2007
Let P(A) be the power set of an n-element set A. Then a(n) = the sum of the size of the symmetric difference of x and y for every subset {x,y} of P(A). - Ross La Haye (rlahaye(AT)new.rr.com), Dec 30 2007
The numbers in this sequence are the Wiener indices of the graphs of n-cubes (boolean hypercubes). For example, the 3-cube is the graph of the standard cube whose Wiener index is 48. - Kailasam Viswanathan Iyer, Feb 26 2009
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2009: (Start)
Starting (1, 8, 48,...) = 4-th binomial transform of [1, 4, 0, 0, 0,...].
Equals the sum of terms in 2^n x 2^n semi magic square arrays in which each row and column
is composed of a binomial frequency of terms in the set (1, 3, 5, 7,...).
The first few such arrays = [1] [1,3; 3,1]; /Q .
[1, 3, 5, 3;
.3, 1, 3, 5;
.5, 3, 1, 3;
.3, 5, 3, 1]
. (sum of terms = 48, with a binomial frequency of (1, 2, 1) as to (1, 3, 5)
in each row and columnn, then
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[1, 3, 5, 3, 5, 7, 5, 3;
.3, 1, 3, 5, 7, 5, 3, 5;
.5, 3, 1, 3, 5, 3, 5, 7;
.3, 5, 3, 1, 3, 5, 7, 5;
.5, 7, 5, 3, 1, 3, 5, 3;
.7, 5, 3, 5, 3, 1, 3, 5;
.5, 3, 5, 7, 5, 3, 1, 3;
.3, 5, 7, 5, 3, 5, 3, 1]
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(in which each row and column is composed of one 1, three 3's, three 5's, and
and one 7; sum of terms in the array = 256.)
... (End)
Sum(n>0,1/a(n)) = 8*log(2) - 4*log(3) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Sep 11 2009]
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REFERENCES
| C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 516.
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. [From Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
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).
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LINKS
| F. Ellermann, Illustration of binomial transforms
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 414
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Wiener Index
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| G.f.: x/(1-4x)^2. A002697(n+1) is the convolution of powers of 4. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
Third binomial transform of n. E.g.f.: x*exp(4x) - Paul Barry (pbarry(AT)wit.ie), Jul 22 2003
a(n)=sum(k=0, n, k*binomial(2*n, 2*k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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)). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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 (pbarry(AT)wit.ie), Oct 15 2004
a(n)=4*a(n-1)+4^(n-1) (with a(0)=0). [From Vincenzo Librandi, Dec 31 2010]
If f(n) = A002697(n+1), then f is the convolution of A000984 with A002457, also the convolution of A000302, f(n) = binomial(n+1,1) * 4^n = (n+1) * 4^n f has G.f.: 1/(1-4x)^2 = (1-4x)^(-2) [From Rui Duarte, Oct 08 2011]
a(0)=0, a(1)=1, a(n)=8*a(n-1)-16*a(n-2) [From Harvey P. Dale, Jan 18 2012]
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MAPLE
| A002697:=1/(4*z-1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Join[{a=0, b=1}, Table[c=8*b-16*a; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 19 2011*)
Table[n 4^(n-1), {n, 0, 30}] (* or *) LinearRecurrence[{8, -16}, {0, 1}, 30] (* From Harvey P. Dale, Jan 18 2012 *)
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PROG
| (PARI) a(n)=if(n<0, 0, n*4^(n-1))
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CROSSREFS
| Cf. A000302, A002697, A027656, A083672, A125145, A128235, A038231, A002699.
Sequence in context: A069021 A079763 A079785 * A026761 A026706 A128734
Adjacent sequences: A002694 A002695 A002696 * A002698 A002699 A002700
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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