OFFSET
0,2
COMMENTS
a(n) = number of Dyck (n+2)-paths with exactly 2 rows of peaks. A row of peaks is a maximal sequence of peaks all at the same height and 2 units apart. For example, UDUDUD ( = /\/\/\ ) contains exactly one row of peaks, as does UUUDDD, but UDUUDDUD has three and a(1)=2 counts UDUUDD, UUDDUD. - David Callan, Mar 02 2005
If X is an n-set and Y a fixed 2-subset of X then a(n-4) is equal to the number of (n-4)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n>=7, a(n-7) is the number of (0,1) n X n matrices A<=P^(-1)+I+P having exactly two 1's in every row and column with perA=16. - Vladimir Shevelev, Apr 12 2010
Row 2 of the convolution array A213550. - Clark Kimberling, Jun 20 2012
a(n-1) = risefac(n, 4)/4! - risefac(n, 2)/2! is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 4 and dimension n. Here risefac is the rising factorial. - Wolfdieter Lang, Dec 10 2015
Consider the array formed by the second polygonal numbers of increasing rank:
A000217(-1-n): 0, 1, 3, 6, 10, 15, ...
A000270(-1-n): 1, 4, 9, 16, 25, 36, ...
A000326(-1-n): 2, 7, 15, 26, 40, 57, ...
A000384(-1-n): 3, 10, 21, 36, 55, 78, ...
Then the antidiagonal sums yield this sequence. - Michael Somos, Nov 23 2021
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.
Vladimir S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=4) (First published: San Francisco: Holden-Day, Inc., 1964)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
Milan Janjic, Two Enumerative Functions
P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380. [See p. 301]
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
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = binomial(n+3, n-1) + binomial(n+2, n-1).
a(n) = binomial(n,4) + 2*binomial(n,3), n>=2. - Zerinvary Lajos, Jul 26 2006
From Colin Barker, Jan 28 2012: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(2-x)/(1-x)^5. (End)
a(n) = Sum_{k=1..n} ( Sum_{i=1..k} i(n-k+2) ). - Wesley Ivan Hurt, Sep 26 2013
a(n+1) = A127672(8+n, n), n >= 0, with the Chebyshev C-polynomial coefficients A127672(n, k). See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015
E.g.f.: (1/24)*x*(48 + 60*x + 16*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 01 2017
Sum_{n>=1} 1/a(n) = 853/1225. - Amiram Eldar, Jan 02 2021
a(n) = A005587(-7-n) for all n in Z. - Michael Somos, Nov 23 2021
MAPLE
[seq(binomial(n, 4)+2*binomial(n, 3), n=2..43)]; # Zerinvary Lajos, Jul 26 2006
seq((n+4)*binomial(n, 4)/n, n=3..43); # Zerinvary Lajos, Feb 28 2007
A005582:=(-2+z)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[n(n+1)(n+2)(n+7)/24, {n, 0, 40}] (* Harvey P. Dale, Jun 01 2012 *)
PROG
(PARI) concat(0, Vec(x*(2-x)/(1-x)^5 + O(x^100))) \\ Altug Alkan, Dec 10 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000
STATUS
approved