A001785 Second-order reciprocal Stirling number (Fekete) a(n) = [[2n+4, n]]. The number of n-orbit permutations of a (2n+4)-set with at least 2 elements in each orbit. Also known as associated Stirling numbers of the first kind (e.g., Comtet). (Formerly M5382 N2338)

1, 120, 7308, 303660, 11098780, 389449060, 13642629000, 486591585480, 17856935296200, 678103775949600, 26726282654771700, 1094862336960892500, 46641683693715610500, 2066075391660447667500, 95122549872697437090000

Offset

0,2

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=0..14.

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+4, n]] = Sum_{i=0..n} (-1)^i*binomial(2n+4, 2n+4-i)*[2n+4-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

Recurrence: 30*(n-1)*(116*n+75)*a(n) + (-6960*n^3-49760*n^2-112691*n-80787)*a(n-1) + (n+1)*(2*n+1)*(20*n+21)*a(n-2) = 0. - R. J. Mathar, Jul 18 2015

For n>0, a(n) = (1113 + 1447*n + 600*n^2 + 80*n^3)*(2*n+4)!/(1215*2^(n+3)*(n-1)!). - Vaclav Kotesovec, Jan 17 2016

Recurrence (for n>1): (n-1)*(80*n^3 + 360*n^2 + 487*n + 186)*a(n) = (n+2)*(2*n+3)*(80*n^3 + 600*n^2 + 1447*n + 1113)*a(n-1). - Vaclav Kotesovec, Jan 18 2016

Maple

with(combinat):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+4, j); od; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

Mathematica

Prepend[Table[Sum[(-1)^i Binomial[2 n + 4, 2 n + 4 - i] Abs@ StirlingS1[2 n + 4 - i, n - i], {i, 0, n}], {n, 14}] , 1] (* Michael De Vlieger, Jan 04 2016 *)

Prog

(PARI) a(n) = if (!n, 1, sum(i=0, n, (-1)^i*binomial(2*n+4, 2*n+4-i)*abs(stirling(2*n+4-i, n-i, 1)))); \\ Michel Marcus, Jan 04 2016
(Magma) [1] cat [(1113+1447*n+600*n^2+80*n^3)*Factorial(2*n+4)/(1215*2^(n+ 3)*Factorial(n-1)): n in [1..15]]; // Vincenzo Librandi, Jan 18 2016

Crossrefs

Cf. A000907, A000483, A001784.

Sequence in context: A035190 A240934 A035815 * A270848 A223960 A351223

Adjacent sequences: A001782 A001783 A001784 * A001786 A001787 A001788

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 0 by Michel Marcus, Jan 04 2016

Status

approved