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%I M4258 N1779 #92 Oct 21 2022 21:57:59
%S 6,50,225,735,1960,4536,9450,18150,32670,55770,91091,143325,218400,
%T 323680,468180,662796,920550,1256850,1689765,2240315,2932776,3795000,
%U 4858750,6160050,7739550,9642906,11921175,14631225,17836160,21605760,26016936,31154200
%N Stirling numbers of first kind, s(n+3, n), negated.
%C 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
%D 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.
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001303/b001303.txt">Table of n, a(n) for n = 1..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Karl Dienger, <a href="/A000217/a000217.pdf">Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung</a>, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
%H G. C. Greubel, <a href="https://arxiv.org/abs/1612.09385">A Note on Jain basis functions</a>, arXiv:1612.09385 [math.CA], 2016.
%H Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., Vol. 1, No. 3 (1926), pp. 44-49. [Annotated scanned copy]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = binomial(n+3, 4)*binomial(n+3, 2).
%F G.f.: x*(6 + 8*x + x^2)/(1 - x)^7. - _Simon Plouffe_ in his 1992 dissertation
%F 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].
%F 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)).
%F From _Jason Lang_, Oct 03 2006: (Start)
%F a(n) = A000217(n) * n! / ( 4! * (n-4)! ) [for n > 4 and A000217 = the triangular numbers];
%F a(n) = ((n+4)! / n! ) ^2 / ( (n+2) * (n+1) * 2*4!);
%F a(n) = (n-0)^2 * (n-1)^2 * (n-2) * (n-3) / (2*4!). (End)
%F From _Miklos Kristof_, Nov 04 2007: (Start)
%F a(n) = 15*binomial(n+5,6) - 10*binomial(n+4,5) + binomial(n+3,4).
%F E.g.f. with offset 4: exp(x)*((1/4)*x^4 + (1/6)*x^5 + (1/48)*x^6). (End)
%F a(n) = n*(n+1)(n+2)^2*(n+3)^2/48. - _Jeremy Galvagni_, Mar 03 2009
%F From _Gary Detlefs_, Jun 06 2010: (Start)
%F a(n) = (n+3)^2/(n^2-1)*a(n-1), n > 1;
%F a(n) = 6*Product_{k=2..n} (k+3)^2/(k^2 - 1). (End)
%F a(n) = A001297(-3-n) for all n in Z. - _Michael Somos_, Sep 04 2017
%F From _Amiram Eldar_, Jan 10 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 16*Pi^2/3 - 472/9.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*Pi^2/3 + 16/9 - 64*log(2)/3. (End)
%p seq(numbperm (n,2)*numbperm (n,4)/48, n=4..33); # _Zerinvary Lajos_, Apr 26 2007
%p seq(15*binomial(n+2,6)-10*binomial(n+1,5)+binomial(n,4),n=4..30); # _Miklos Kristof_, Nov 04 2007
%p A001303 := proc(n)
%p -combinat[stirling1](n+3,n) ;
%p end proc: # _R. J. Mathar_, May 19 2016
%t Table[-StirlingS1[n + 3, n], {n, 100}] (* _T. D. Noe_, Jun 27 2012 *)
%t a[ n_] := n (n + 1) (n + 2)^2 (n + 3)^2 / 48; (* _Michael Somos_, Sep 04 2017 *)
%o (Sage) [stirling_number1(n,n-3) for n in range(4, 34)] # _Zerinvary Lajos_, May 16 2009
%o (PARI) a(n) = n*(n+1)*(n+2)^2*(n+3)^2/48; \\ _Altug Alkan_, Aug 29 2017
%Y Cf. A000217, A001297, A008275.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
%E Notation of the polynomial formula edited by _R. J. Mathar_, Sep 15 2009
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