OFFSET
1,1
COMMENTS
If X is an n-set and Y a fixed 2-subset of X then a(n-5) is equal to the number of (n-5)-subsets of X intersecting Y. -Milan Janjic,Jul 30 2007
a(n-1) = risefac(n,5)/5! - risefac(n,3)/3! is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 5 and dimension n. Here risefac is the rising factorial. Put a(0) = 0. -Wolfdieter Lang,Dec 10 2015
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
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 = 1..172
Milan Janjic,Two Enumerative Functions.
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].
Richard K. Guy,Letter to N. J. A. Sloane, Feb 1988.
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.
Cecilia Rossiter,Depictions, Explorations and Formulas of the Euler/Pascal Cube.[broken link]
Cecilia Rossiter,Depictions, Explorations and Formulas of the Euler/Pascal Cube.[Cached copy, May 15 2013]
FORMULA
G.f.: x*(2-x)/(1-x)^6.
a(n) = binomial(n+4, n-1) + binomial(n+3, n-1) = (1/120)*n*(n+9)*(n+3)*(n+2)*(n+1).
a(n+1) = -A127672(10+n, n), n >= 0, with the coefficients of the Chebyshev C-polynomialsA127672(n, k). -Wolfdieter Lang,Dec 10 2015
a(n) = binomial(n+3,5) + 2*binomial(n+3,4). -Yuchun Ji,May 23 2019
FromAmiram Eldar,Feb 17 2023: (Start)
Sum_{n>=1} 1/a(n) = 40751/63504.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1360*log(2)/63 - 922961/63504. (End)
MAPLE
A005583:=-(-2+z)/(z-1)**6; #Simon Plouffein his 1992 dissertation (this g.f. assumes offset 0)
PROG
(PARI)
conv(u, v)=local(w); w=vector(length(u), i, sum(j=1, i, u[j]*v[i+1-j])); w;
t(n)=n*(n+1)/2;
u=vector(10, i, t(i));
v=vector(10, i, t(i)-1);
conv(u, v)
(PARI) a(n) = (1/120)*n*(n+9)*(n+3)*(n+2)*(n+1); \\Joerg Arndt,Mar 05 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 07 1999
More terms fromZerinvary Lajos,Jul 21 2006
STATUS
approved