A003472 a(n) = 2^(n-4)*C(n,4).

1, 10, 60, 280, 1120, 4032, 13440, 42240, 126720, 366080, 1025024, 2795520, 7454720, 19496960, 50135040, 127008768, 317521920, 784465920, 1917583360, 4642570240, 11142168576, 26528972800, 62704844800, 147220070400

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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).

Crossrefs

Cf. A001787, A001788, A001789.

a(n) = A038207(n,4).

Programs

GAP

List([4..30], n-> 2^(n-4)*Binomial(n,4)); # G. C. Greubel, Aug 27 2019

Magma

[2^(n-4)*Binomial(n, 4): n in [4..30]]; // Vincenzo Librandi, Oct 16 2011

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 *)

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

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)

Extensions

More terms from James Sellers, Apr 15 2000