OFFSET
0,1
REFERENCES
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 = 0..1000
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
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 (6,-15,20,-15,6,-1).
FORMULA
a(n) = A009766(n+5,5) = (n+1)*binomial(n+10,4)/5.
G.f.: (42 - 120*x + 135*x^2 - 70*x^3 + 14*x^4)/(1-x)^6; numerator polynomial is N(2;4, x) from A062991.
a(n) = binomial(n+9,5) - binomial(n+9,3). - Zerinvary Lajos, Jul 19 2006
a(n) = A214292(n+9,4). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 2509/63504.
Sum_{n>=0} (-1)^n/a(n) = 951395/63504 - 1360*log(2)/63. (End)
From Elmo R. Oliveira, Sep 09 2025: (Start)
a(n) = (n + 1)*(n + 7)*(n + 8)*(n + 9)*(n + 10)/120.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
E.g.f.: exp(x)*(5040 + 10800*x + 4500*x^2 + 700*x^3 + 45*x^4 + x^5)/120. (End)
MAPLE
[seq(binomial(n, 5)-binomial(n, 3), n=9..55)]; # Zerinvary Lajos, Jul 19 2006
A005557:=(42-120*z+135*z**2-70*z**3+14*z**4)#(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
CoefficientList[Series[(14 z^4 - 70 z^3 + 135 z^2 - 120 z + 42)/(z - 1)^6, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {42, 132, 297, 572, 1001, 1638}, 40] (* Harvey P. Dale, Feb 22 2024 *)
PROG
(Magma) [(n+1)*Binomial(n+10, 4)/5: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013
(GAP) List([0..30], n->(n+1)*Binomial(n+10, 4)/5); # Muniru A Asiru, Apr 10 2018
(PARI) a(n)=(n+1)*binomial(n+10, 4)/5 \\ Charles R Greathouse IV, Oct 21 2022
CROSSREFS
KEYWORD
nonn,walk,easy
AUTHOR
EXTENSIONS
More terms and formula from Wolfdieter Lang, Sep 04 2001
Name clarified by Alois P. Heinz, Sep 09 2025
STATUS
approved
