A000907 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). (Formerly M4298 N1797)

6, 130, 2380, 44100, 866250, 18288270, 416215800, 10199989800, 268438920750, 7562120816250, 227266937597700, 7262844156067500, 246045975136211250, 8810836639999143750, 332624558868351750000, 13205706717164131170000

Offset

1,1

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.

C. Jordan, Calculus of Finite Differences. Budapest, 1939, p. 152.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..16.

A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

H. W. Gould, Harris Kwong, and Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.

C. Jordan, On Stirling's Numbers, Tohoku Math. J., 37 (1933), 254-278.

Formula

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

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

a(n) = (4*n+5)*(2*n+2)!/(9*2^(n+1)*(n-1)!). - Vaclav Kotesovec, Jan 17 2016

Maple

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000483, A001784, A001785.

Sequence in context: A318528 A095695 A156475 * A188718 A077031 A302770

Adjacent sequences: A000904 A000905 A000906 * A000908 A000909 A000910

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Offset changed to 1 by Michel Marcus, Jan 04 2016

Status

approved