A001784 -id:A001784 - cp's OEIS frontend

A000483 Associated Stirling numbers: second-order reciprocal Stirling numbers (Fekete) a(n) = [[n, 3]]. The number of 3-orbit permutations of an n-set with at least 2 elements in each orbit.

15, 210, 2380, 26432, 303660, 3678840, 47324376, 647536032, 9418945536, 145410580224, 2377609752960, 41082721413120, 748459539843840, 14345340443665920, 288650580508961280, 6085390148673177600, 134167064248901376000, 3088040233895705088000, 74077507611407752704000, 1849221425299053367296000

Offset: 6

N. J. A. Sloane

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.
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).

Alois P. Heinz, Table of n, a(n) for n = 6..200
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

Cf. A000907, A001784, A001785. A diagonal of triangle in A008306.

Cf. A000276.

nn=25;a=Log[1/(1-x)]-x;Drop[Range[0,nn]!CoefficientList[Series[a^3/3!, {x,0,nn}],x],6]  (* Geoffrey Critzer, Nov 03 2012 *)

With alternating signs: Ramanujan polynomials psi_4(n-3, x) evaluated at 1. - Ralf Stephan, Apr 16 2004
E.g.f.: -((x+log(1-x))^3)/6. - Vladeta Jovovic, May 03 2008
Conjecture: (n-2)*(n-4)*a(n) -(n-1)*(3*n^2-21*n+35)*a(n-1) +(n-1)*(n-2)*(3*n^2-24*n+47)*a(n-2) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 18 2015
Conjecture: 3*(-n+4)*a(n) +(9*n^2-59*n+90)*a(n-1) +(-9*n^3+96*n^2-348*n+436)*a(n-2) +(n-3)*(3*n^3-45*n^2+237*n-430)*a(n-3) +5*(n-5)*(n-6)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 18 2015

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

More terms from Sean A. Irvine, Nov 14 2010

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).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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).

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.

Cf. A000483, A001784, A001785.

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

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 *)

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

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

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

Offset changed to 1 by Michel Marcus, Jan 04 2016

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).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

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).

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.

Cf. A000907, A000483, A001784.

[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

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

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 *)

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

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

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

Offset changed to 0 by Michel Marcus, Jan 04 2016

cp's OEIS Frontend

A000483 Associated Stirling numbers: second-order reciprocal Stirling numbers (Fekete) a(n) = [[n, 3]]. The number of 3-orbit permutations of an n-set with at least 2 elements in each orbit.

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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).

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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).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions