A000907 - OEIS

%I M4298 N1797 #50 Aug 05 2023 13:08:12

%S 6,130,2380,44100,866250,18288270,416215800,10199989800,268438920750,

%T 7562120816250,227266937597700,7262844156067500,246045975136211250,

%U 8810836639999143750,332624558868351750000,13205706717164131170000

%N Second-order reciprocal Stirling number (Fekete) a(n) = [[2n+2, n]]. The number of n-orbit permutations of a (2n+2)-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+2, n]] = Sum_{i=0..n} (-1)^i*binomial(2n+2, 2n+2-i)*[2n+2-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: n*(4*n+5)*a(n) -(2*n+3)*(n+2)*(4*n+9)*a(n-1)=0. - _R. J. Mathar_, Apr 30 2015

%F a(n) = (4*n+5)*(2*n+2)!/(9*2^(n+1)*(n-1)!). - _Vaclav Kotesovec_, Jan 17 2016

%p s1 := (n,k)->sum((-1)^i*binomial(n,i)*abs(stirling1(n-i,k-i)),i=0..n); for j from 1 to 20 do s1(2*j+2,j); od; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

%t Table[Sum[(-1)^i Binomial[2 n + 2, 2 n + 2 - i] Abs@ StirlingS1[2 n + 2 - i, n - i], {i, 0, n}], {n, 16}] (* _Michael De Vlieger_, Jan 04 2016 *)

%o (PARI) a(n) = sum(i=0, n, (-1)^i*binomial(2*n+2, 2*n+2-i)*abs(stirling(2*n+2-i, n-i, 1))); \\ _Michel Marcus_, Jan 04 2016

%Y Cf. A000483, A001784, A001785.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

%E Offset changed to 1 by _Michel Marcus_, Jan 04 2016