OFFSET
1,1
COMMENTS
a(n) is equal to the sum of the products of each distinct grouping of 3 members of the set {1, 2, 3, ..., n + 2} (a(1) = 1*2*3, a(2) = 1*2*3 + 1*2*4 + 1*3*4 + 2*3*4, a(3) = 1*2*3 + 1*2*4 + 1*2*5 + 1*3*4 + 1*3*5 + 1*4*5 + 2*3*4 + 2*3*5 + 2*4*5 + 3*4*5). - Jeffreylee R. Snow, Sep 23 2013
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. 833.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
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].
Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
G. C. Greubel, A Note on Jain basis functions, arXiv:1612.09385 [math.CA], 2016.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., Vol. 1, No. 3 (1926), pp. 44-49. [Annotated scanned copy]
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 (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = binomial(n+3, 4)*binomial(n+3, 2).
G.f.: x*(6 + 8*x + x^2)/(1 - x)^7. - Simon Plouffe in his 1992 dissertation
E.g.f. with offset 3: exp(x)*(6*(x^3)/3! + 26*(x^4)/4! + 35*(x^5)/5! + 15*(x^6)/6!). See row k=3 of A112486 for the coefficients [6, 26, 35, 15].
a(n) = (f(n+2, 3)/6!)*Sum_{m=0..min(3, n)} A112486(3,m)*f(6, 3-m)*f(n-1, m), with the falling factorials notation f(n, m):=n*(n-1)*...*(n-(m-1)).
From Jason Lang, Oct 03 2006: (Start)
a(n) = ((n+4)! / n! ) ^2 / ( (n+2) * (n+1) * 2*4!);
a(n) = (n-0)^2 * (n-1)^2 * (n-2) * (n-3) / (2*4!). (End)
From Miklos Kristof, Nov 04 2007: (Start)
a(n) = 15*binomial(n+5,6) - 10*binomial(n+4,5) + binomial(n+3,4).
E.g.f. with offset 4: exp(x)*((1/4)*x^4 + (1/6)*x^5 + (1/48)*x^6). (End)
a(n) = n*(n+1)(n+2)^2*(n+3)^2/48. - Jeremy Galvagni, Mar 03 2009
From Gary Detlefs, Jun 06 2010: (Start)
a(n) = (n+3)^2/(n^2-1)*a(n-1), n > 1;
a(n) = 6*Product_{k=2..n} (k+3)^2/(k^2 - 1). (End)
a(n) = A001297(-3-n) for all n in Z. - Michael Somos, Sep 04 2017
From Amiram Eldar, Jan 10 2022: (Start)
Sum_{n>=1} 1/a(n) = 16*Pi^2/3 - 472/9.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*Pi^2/3 + 16/9 - 64*log(2)/3. (End)
MAPLE
seq(numbperm (n, 2)*numbperm (n, 4)/48, n=4..33); # Zerinvary Lajos, Apr 26 2007
seq(15*binomial(n+2, 6)-10*binomial(n+1, 5)+binomial(n, 4), n=4..30); # Miklos Kristof, Nov 04 2007
A001303 := proc(n)
-combinat[stirling1](n+3, n) ;
end proc: # R. J. Mathar, May 19 2016
MATHEMATICA
Table[-StirlingS1[n + 3, n], {n, 100}] (* T. D. Noe, Jun 27 2012 *)
a[ n_] := n (n + 1) (n + 2)^2 (n + 3)^2 / 48; (* Michael Somos, Sep 04 2017 *)
PROG
(Sage) [stirling_number1(n, n-3) for n in range(4, 34)] # Zerinvary Lajos, May 16 2009
(PARI) a(n) = n*(n+1)*(n+2)^2*(n+3)^2/48; \\ Altug Alkan, Aug 29 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
Notation of the polynomial formula edited by R. J. Mathar, Sep 15 2009
STATUS
approved