|
| |
|
|
A003472
|
|
a(n) = 2^(n-4)*C(n,4).
(Formerly M4718)
|
|
26
|
|
|
|
1, 10, 60, 280, 1120, 4032, 13440, 42240, 126720, 366080, 1025024, 2795520, 7454720, 19496960, 50135040, 127008768, 317521920, 784465920, 1917583360, 4642570240, 11142168576, 26528972800, 62704844800, 147220070400
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
4,2
|
|
|
COMMENTS
|
Number of 4D hypercubes in n-dimensional hypercube. - Henry Bottomley, Apr 14 2000
With four leading zeros, binomial transform of C(n,4). - Paul Barry, Apr 10 2003
If X_1, X_2, ..., X_n is a partition of a 2n-set X into 2-blocks, then, for n>3, a(n) is equal to the number of (n+4)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
|
|
|
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.
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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
|
Vincenzo Librandi, Table of n, a(n) for n = 4..400
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].
H. J. Brothers, Pascal's Prism: Supplementary Material.
Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
Herbert Izbicki, Über Unterbaeume eines Baumes, Monatshefte für Mathematik, Vol. 74 (1970), pp. 56-62.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
C. W. Jones, J. C. P. Miller, J. F. C. Conn and R. C. Pankhurst, Tables of Chebyshev polynomials Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
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.
John Riordan and N. J. A. Sloane, Correspondence, 1974.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).
|
|
|
FORMULA
|
a(n) = 2*a(n-1) + A001789(n-1).
From Paul Barry, Apr 10 2003: (Start)
O.g.f.: x^4/(1-2*x)^5.
E.g.f.: exp(2*x)(x^4/4!) (with 4 leading zeros). (End)
a(n) = Sum_{i=4..n} binomial(i,4)*binomial(n,i). Example: for n=7, a(7) = 1*35 + 5*21 + 15*7 + 35*1 = 280. - Bruno Berselli, Mar 23 2018
From Amiram Eldar, Jan 06 2022: (Start)
Sum_{n>=4} 1/a(n) = 20/3 - 8*log(2).
Sum_{n>=4} (-1)^n/a(n) = 216*log(3/2) - 260/3. (End)
|
|
|
MAPLE
|
A003472:=-1/(2*z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation
seq(binomial(n, 4)*2^(n-4), n=4..24); # Zerinvary Lajos, Jun 12 2008
|
|
|
MATHEMATICA
|
Table[2^(n-4) Binomial[n, 4], {n, 4, 50}] (* or *) LinearRecurrence[{10, -40, 80, -80, 32}, {1, 10, 60, 280, 1120}, 50] (* Harvey P. Dale, May 27 2017 *)
|
|
|
PROG
|
(MAGMA) [2^(n-4)*Binomial(n, 4): n in [4..30]]; // Vincenzo Librandi, Oct 16 2011
(PARI) a(n)=binomial(n, 4)<<(n-4) \\ Charles R Greathouse IV, May 18 2015
(Sage) [2^(n-4)*binomial(n, 4) for n in (4..30)] # G. C. Greubel, Aug 27 2019
(GAP) List([4..30], n-> 2^(n-4)*Binomial(n, 4)); # G. C. Greubel, Aug 27 2019
|
|
|
CROSSREFS
|
Cf. A001787, A001788, A001789.
a(n) = A038207(n,4).
Sequence in context: A278721 A341366 A004406 * A112502 A293081 A292489
Adjacent sequences: A003469 A003470 A003471 * A003473 A003474 A003475
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane
|
|
|
EXTENSIONS
|
More terms from James A. Sellers, Apr 15 2000
|
|
|
STATUS
|
approved
|
| |
|
|
