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A037960 a(n) = n*(3*n+1)*(n+2)!/24.

%I A037960 #38 Dec 07 2024 02:02:17
%S A037960 0,1,14,150,1560,16800,191520,2328480,30240000,419126400,6187104000,
%T A037960 97037740800,1612798387200,28332944640000,524813313024000,
%U A037960 10226013557760000,209144207720448000,4480594531725312000,100357207837286400000,2345925761384325120000,57136703662028390400000
%N A037960 a(n) = n*(3*n+1)*(n+2)!/24.
%C A037960 For n>=1, a(n) is equal to the number of surjections from {1,2,..,n+2} onto {1,2,...,n}. - Aleksandar M. Janjic and _Milan Janjic_, Feb 24 2007
%D A037960 Identity (1.18) in H. W. Gould, Combinatorial Identities, Morgantown, 1972; page 3.
%H A037960 Vincenzo Librandi, <a href="/A037960/b037960.txt">Table of n, a(n) for n = 0..300</a>
%H A037960 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A037960 H. W. Gould, ed. J. Quaintance, <a href="http://www.math.wvu.edu/~hgould/Vol.4.PDF">Combinatorial Identities</a>, May 2010 (identity 10.3, p.45)
%F A037960 a(n) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - _Vladimir Kruchinin_, Jun 01 2013
%F A037960 (3*n-2)*(n-1)*a(n) - n*(n+2)*(3*n+1)*a(n-1) = 0. - _R. J. Mathar_, Jul 26 2015
%F A037960 E.g.f.: x*(1 + 2*x)/(1 - x)^5. - _Ilya Gutkovskiy_, Feb 20 2017
%F A037960 From _G. C. Greubel_, Jun 20 2022: (Start)
%F A037960 a(n) = n!*StirlingS2(n+2, n).
%F A037960 a(n) = A131689(n+2, n).
%F A037960 a(n) = A019538(n+2, n). (End)
%t A037960 Table[(n+2)!*n*(3n+1)/24,{n,0,20}] (* _Harvey P. Dale_, Oct 16 2014 *)
%o A037960 (PARI) n*(3*n+1)*(n+2)!/24 \\ _Charles R Greathouse IV_, Nov 02 2011
%o A037960 (Magma) [Factorial(n+2)*n*(3*n+1)/24: n in [0..25]]; // _Vincenzo Librandi_, Feb 20 2017
%o A037960 (SageMath) [factorial(n)*stirling_number2(n+2, n) for n in (0..30)] # _G. C. Greubel_, Jun 20 2022
%Y A037960 Cf. A000142, A001286, A001296, A019538, A037960, A131689.
%Y A037960 Cf. A037959, A037961, A037962, A037963.
%K A037960 nonn,easy
%O A037960 0,3
%A A037960 _N. J. A. Sloane_
%E A037960 More terms from _Vincenzo Librandi_, Feb 20 2017