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A001702 Generalized Stirling numbers.

%I A001702 M5148 N2234 #63 Sep 08 2022 08:44:29
%S A001702 1,24,154,580,1665,4025,8624,16884,30810,53130,87450,138424,211939,
%T A001702 315315,457520,649400,903924,1236444,1664970,2210460,2897125,3752749,
%U A001702 4809024,6101900,7671950,9564750,11831274,14528304,17718855,21472615,25866400,30984624
%N A001702 Generalized Stirling numbers.
%D A001702 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001702 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001702 T. D. Noe, <a href="/A001702/b001702.txt">Table of n, a(n) for n = 1..1000</a>
%H A001702 D. S. Mitrinovic and R. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
%H A001702 D. S. Mitrinovic and R. S. Mitrinovic, <a href="https://www.jstor.org/stable/43667130">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
%H A001702 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 A001702 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 A001702 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001702 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A001702 a(n) = (1/48)*(n-1)*n*(n+1)*(n+4)*(n^2+7n+14), n > 1.
%F A001702 G.f.: x + x^2*(x-4)*(x^2-2*x+6)/(x-1)^7. - _Simon Plouffe_ in his 1992 dissertation
%F A001702 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
%F A001702 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - _Colin Barker_, Jul 08 2020
%p A001702 A001702 := proc(n)
%p A001702     if n = 1 then
%p A001702         1 ;
%p A001702     else
%p A001702         (n-1)*n*(n+1)*(n+4)*(n^2+7*n+14)/48 ;
%p A001702     end if;
%p A001702 end proc: # _R. J. Mathar_, Sep 23 2016
%t A001702 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 A001702 CoefficientList[Series[1 +x*(x-4)*(x^2-2*x+6)/(x-1)^7, {x, 0, 100}], x] (* _Stefano Spezia_, Sep 30 2018 *)
%t A001702 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 A001702 (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 A001702 (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 A001702 (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 A001702 For n > 1, a(n) = A145324(n+2,4).
%K A001702 nonn,easy
%O A001702 1,2
%A A001702 _N. J. A. Sloane_