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A001784 Second order reciprocal Stirling number (Fekete) [[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).
(Formerly M5169 N2244)
3
1, 24, 924, 26432, 705320, 18858840, 520059540, 14980405440, 453247114320, 14433720701400, 483908513388300, 17068210823664000, 632607429473019000, 24602295329058447000, 1002393959071727722500, 42720592574082543120000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

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

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..15.

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

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

FORMULA

[[2n+3, n]]=sum((-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

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

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

For n>0, a(n) = (67 + 75*n + 20*n^2)*(2*n+3)!/(405*2^n*(n-1)!). - Vaclav Kotesovec, Jan 17 2016

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A000907, A000483, A001785.

Sequence in context: A006147 A061236 A266997 * A172206 A220804 A220253

Adjacent sequences:  A001781 A001782 A001783 * A001785 A001786 A001787

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

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Last modified October 16 13:32 EDT 2019. Contains 328093 sequences. (Running on oeis4.)