%I M5169 N2244 #52 Aug 05 2023 13:08:32
%S 1,24,924,26432,705320,18858840,520059540,14980405440,453247114320,
%T 14433720701400,483908513388300,17068210823664000,632607429473019000,
%U 24602295329058447000,1002393959071727722500,42720592574082543120000
%N Second-order reciprocal Stirling number (Fekete) a(n) = [[2n+3, n]]. The number of n-orbit permutations of a (2n+3)-set with at least 2 elements in each orbit. Also known as associated Stirling numbers of the first kind (e.g., Comtet).
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
%D C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 152.
%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 A. E. Fekete, <a href="http://www.jstor.org/stable/2974533">Apropos two notes on notation</a>, Amer. Math. Monthly, 101 (1994), 771-778.
%H H. W. Gould, Harris Kwong, and Jocelyn Quaintance, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kwong/kwong9.html">On Certain Sums of Stirling Numbers with Binomial Coefficients</a>, J. Integer Sequences, 18 (2015), #15.9.6.
%H C. Jordan, <a href="https://www.jstage.jst.go.jp/article/tmj1911/37/0/37_0_254/_pdf">On Stirling's Numbers</a>, Tohoku Math. J., 37 (1933), 254-278.
%F a(n) = [[2n+3, n]] = Sum_{i=0..n} (-1)^i*binomial(2n+3, 2n+3-i)*[2n+3-i, n-i] where [n, k] is the unsigned Stirling number of the first kind. - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
%F Conjecture: 480*(n+1)*a(n) +30*(-32*n^2-14821*n+42287)*a(n-1) +(878700*n^2-403433*n+5134227)*a(n-2) +(911423*n-656446)*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Jul 18 2015
%F Conjecture: (n-2)*(20*n^2-5*n-3)*a(n) -n*(2*n+1)*(20*n^2+35*n+12)*a(n-1)=0. - _R. J. Mathar_, Jul 18 2015
%F For n>0, a(n) = (67 + 75*n + 20*n^2)*(2*n+3)!/(405*2^n*(n-1)!). - _Vaclav Kotesovec_, Jan 17 2016
%p with(combinat):s1 := (n,k)->sum((-1)^i*binomial(n,i)*abs(stirling1(n-i,k-i)),i=0..n); 1; for j from 1 to 20 do s1(2*j+3,j); od; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
%t Prepend[Table[Sum[(-1)^i Binomial[2 n + 3, 2 n + 3 - i] Abs@ StirlingS1[2 n + 3 - i, n - i], {i, 0, n}], {n, 15}] , 1] (* _Michael De Vlieger_, Jan 04 2016 *)
%o (PARI) a(n) = if (!n, 1, sum(i=0, n, (-1)^i*binomial(2*n+3, 2*n+3-i)*abs(stirling(2*n+3-i, n-i, 1)))); \\ _Michel Marcus_, Jan 04 2016
%Y Cf. A000907, A000483, A001785.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
%E Offset changed to 0 by _Michel Marcus_, Jan 04 2016