cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

15, 210, 2380, 26432, 303660, 3678840, 47324376, 647536032, 9418945536, 145410580224, 2377609752960, 41082721413120, 748459539843840, 14345340443665920, 288650580508961280, 6085390148673177600, 134167064248901376000, 3088040233895705088000, 74077507611407752704000, 1849221425299053367296000
Offset: 6

Views

Author

N. J. A. Sloane

Keywords

References

  • 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).

Links

Crossrefs

Cf. A000907, A001784, A001785. A diagonal of triangle in A008306.
Cf. A000276.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
More terms from Sean A. Irvine, Nov 14 2010

A001784 Second-order reciprocal Stirling number (Fekete) a(n) = [[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).

Original entry on oeis.org

1, 24, 924, 26432, 705320, 18858840, 520059540, 14980405440, 453247114320, 14433720701400, 483908513388300, 17068210823664000, 632607429473019000, 24602295329058447000, 1002393959071727722500, 42720592574082543120000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

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

Crossrefs

Cf. A000907, A000483, A001785.

Programs

  • 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 *)
  • 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

Formula

a(n) = [[2n+3, n]] = Sum_{i=0..n} (-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

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

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

Views

Author

N. J. A. Sloane

Keywords

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

Crossrefs

Cf. A000907, A000483, A001784.

Programs

  • 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
  • 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 *)
  • 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

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

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
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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