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A001789
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Binomial(n,3)*2^(n-3).
(Formerly M4522 N1916)
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22
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1, 8, 40, 160, 560, 1792, 5376, 15360, 42240, 112640, 292864, 745472, 1863680, 4587520, 11141120, 26738688, 63504384, 149422080, 348651520, 807403520, 1857028096, 4244635648, 9646899200, 21810380800, 49073356800, 109924319232
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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COMMENTS
| Number of 3-dimensional cubes in n-dimensional hypercube - Henry Bottomley (se16(AT)btinternet.com), Apr 14 2000
With three leading zeros, this is the second binomial transform of (0,0,0,1,0,0,0,0,..) - Paul Barry (pbarry(AT)wit.ie), Mar 07 2003
With 3 leading zeros, binomial transform of C(n,3). - Paul Barry (pbarry(AT)wit.ie), Apr 10 2003
Let M=[1,0,i;0,1,0;i,0,1], i=sqrt(-1). Then 1/det(I-xM)=1/(1-2x)^4. - Paul Barry (pbarry(AT)wit.ie), Apr 27 2005
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>2, a(n+1) is equal to the number of (n+3)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan R. Janjic (agnus(AT)blic.net), Jul 21 2007
With a different offset, number of n-permutations (n=4) of 3 objects: u, v, w with repetition allowed, containing exactly three u's. Example: a(1)=8 because we have: uuuv, uuvu, uvuu, vuuu, uuuw, uuwu, uwuu and wuuu - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2008
With offset 0, a(n) is the number of ways to seperate [n] into four non-overlapping intervals (allowed to be empty) and then choose a subset from each interval. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443), Feb 07 2009]
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
H. J. Brothers, Pascal's Prism: Supplementary Material, http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf.
H. Izbicki, Ueber Unterbaeume eines Baumes, Monatshefte f\"{u}r Mathematik, 74 (1970), 56-62.
Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946). 187-203.
Dusko Letic, Nenad Cakic, Branko Davidovic, Ivana Berkovic and Eleonora Desnica, Some certain properties of the generalized hypercubical functions, Advances in Difference Equations,
2011, 2011:60; http://www.advancesindifferenceequations.com/content/pdf/1687-1847-2011-60.pdf.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
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
| T. D. Noe, Table of n, a(n) for n=3..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjic, Two Enumerative Functions
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, Hypercube
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n)=2*a(n-1)+A001788(n-1)
G.f. (with three leading zeros): x^3/(1-2x)^4. With three leading zeros, a(n)=8a(n-1)-24a(n-2)+32a(n-3)-16a(n-4), a(0)=a(1)=a(2)=0, a(3)=1. - Paul Barry (pbarry(AT)wit.ie), Mar 07 2003
E.g.f. (x^3/3!)exp(2x) (with 3 leading zeros) - Paul Barry (pbarry(AT)wit.ie), Apr 10 2003
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MAPLE
| seq((n^3-n)*2^(n-3)/3, n=2..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007
A001789:=1/(2*z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
seq(binomial(n+3, 3)*2^n, n=0..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2008
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MATHEMATICA
| Table[Binomial[n, 3]*2^(n - 3), {n, 3, 100}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 18 2006
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PROG
| (Other) SAGE: [lucas_number1(n, 2, 0)*binomial(n, 3)/4 for n in xrange(3, 29)] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 10 2009]
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CROSSREFS
| Cf. A001787, A001788, A003472.
For n>0, a(n+3) = 2 * A082138(n) = 8 * A080930(n+1).
Sequence in context: A125198 A128639 A004405 * A074412 A113071 A006726
Adjacent sequences: A001786 A001787 A001788 * A001790 A001791 A001792
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 15 2000
More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 18 2006
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