%I M4540 N1927 #63 Jan 30 2024 21:21:05
%S 0,1,8,54,384,3000,25920,246960,2580480,29393280,362880000,4829932800,
%T 68976230400,1052366515200,17086945075200,294226732800000,
%U 5356234211328000,102793666719744000,2074369080655872000,43913881247588352000,973160803270656000000,22531105497723863040000
%N a(n) = n^2 * n!.
%C Denominators in power series expansion of the higher order exponential integral E(x,m=2,n=1) - (gamma^2/2 + Pi^2/12 + gamma*log(x) + log(x)^2/2), n>0, see A163931. - _Johannes W. Meijer_, Oct 16 2009
%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 Delbert L. Johnson, <a href="/A002775/b002775.txt">Table of n, a(n) for n = 0..447</a>
%H J. D. H. Dickson, <a href="https://doi.org/10.1112/plms/s1-10.1.120">Discussion of two double series arising from the number of terms in determinants of certain forms</a>, Proc. London Math. Soc., 10 (1879), 120-122.
%H J. D. H. Dickson, <a href="/A002775/a002775.pdf">Discussion of two double series arising from the number of terms in determinants of certain forms</a>, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]
%F E.g.f.: x*(1+x)/(1-x)^3. - _Vladeta Jovovic_, Dec 01 2002
%F E.g.f.: x*A'(x) where A(x) is the e.g.f. for A001563. - _Geoffrey Critzer_, Jan 17 2012
%F From _Alexander Adamchuk_, Oct 24 2004: (Start)
%F Sum of all matrix elements M(i, j) = i/(i+j) multiplied by 2*n!.
%F a(n) = 2 * n! * Sum_{j=1..n} Sum_{i=1..n} i/(i+j).
%F Example: a(2) = 2*2! * (1/(1+1) + 1/(1+2) + 2/(2+1) + 2/(2+2)) = 8. (End)
%F From _Amiram Eldar_, Dec 24 2023: (Start)
%F Sum_{n>=1} 1/a(n) = A367731.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = A367732. (End)
%p with(combinat):for n from 0 to 15 do printf(`%d, `,n!/2*sum(2*n, k=1..n)) od: # _Zerinvary Lajos_, Mar 13 2007
%p seq(sum(sum(mul(k, k=1..n),l=1..n),m=1..n), n=0..21); # _Zerinvary Lajos_, Jan 26 2008
%p with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*n^2, n=0..21); # _Zerinvary Lajos_, Jun 11 2008
%p a:=n->add(0+add(n!, j=1..n),j=1..n):seq(a(n), n=0..21); # _Zerinvary Lajos_, Aug 27 2008
%t nn=20;a=1/(1-x); Range[0,nn]! CoefficientList[Series[x D[x D[a,x], x], {x,0,nn}], x] (* _Geoffrey Critzer_, Jan 17 2012 *)
%t Table[n^2 n!,{n,0,40}] (* _Harvey P. Dale_, Aug 01 2021 *)
%Y Cf. A000290, A000142, A047922.
%Y Cf. A163931 (E(x,m,n)), A001563 (n*n!), A091363 (n^3*n!), A091364 (n^4*n!). - _Johannes W. Meijer_, Oct 16 2009
%Y Cf. A367731, A367732.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_