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A001702 Generalized Stirling numbers.
(Formerly M5148 N2234)
2

%I M5148 N2234

%S 1,24,154,580,1665,4025,8624,16884,30810,53130,87450,138424,211939,

%T 315315,457520,649400,903924,1236444,1664970,2210460,2897125,3752749,

%U 4809024,6101900,7671950,9564750,11831274,14528304,17718855,21472615,25866400,30984624

%N Generalized Stirling numbers.

%D Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres reliés aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

%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="/A001702/b001702.txt">Table of n, a(n) for n = 1..1000</a>

%H Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%F a(n) = (1/48)*(n-1)*n*(n+1)*(n+4)*(n^2+7n+14), n > 1.

%F G.f.: x + x^2*(x-4)*(x^2-2*x+6)/(x-1)^7. - _Simon Plouffe_ in his 1992 dissertation

%F If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a - j), then a(n-1) = -f(n,n-3,2), for n >= 3. - _Milan Janjic_, Dec 20 2008

%p A001702 := proc(n)

%p if n = 1 then

%p 1 ;

%p else

%p (n-1)*n*(n+1)*(n+4)*(n^2+7*n+14)/48 ;

%p end if;

%p end proc: # _R. J. Mathar_, Sep 23 2016

%t Join[{1}, Table[(n-1) n (n+1) (n+4) (n^2 + 7 n + 14)/48, {n, 2, 100}]] (* _T. D. Noe_, Aug 09 2012 *)

%t CoefficientList[Series[1 +x*(x-4)*(x^2-2*x+6)/(x-1)^7, {x, 0, 100}], x] (* _Stefano Spezia_, Sep 30 2018 *)

%t Join[{1},Table[Coefficient[Product[x + j, {j, 2, k}], x, k - 4], {k, 4, 40}]] (* or *) Join[{1}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {24, 154, 580, 1665, 4025, 8624, 16884}, 40]] (* _Robert A. Russell_, Oct 04 2018 *)

%o (GAP) Concatenation([1],List([2..35],n->(n-1)*n*(n+1)*(n+4)*(n^2+7*n+14)/48)); # _Muniru A Asiru_, Sep 29 2018

%o (MAGMA) [1] cat [n*(n^2-1)*(n+4)*(n^2+7*n+14)/48: n in [2..35]]; // _Vincenzo Librandi_, Sep 30 2018

%o (PARI) vector(50, n, if(n==1, 1, (1/48)*(n-1)*n* (n+1)* (n+4)*(n^2 +7*n +14))) \\_G. C. Greubel_, Oct 06 2018

%Y For n > 1, a(n) = A145324(n+2,4).

%K nonn

%O 1,2

%A _N. J. A. Sloane_

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Last modified March 18 19:58 EDT 2019. Contains 321293 sequences. (Running on oeis4.)