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 1-5 of 5 results.

A001277 Number of permutations of length n by rises.

Original entry on oeis.org

1, 3, 12, 56, 321, 2175, 17008, 150504, 1485465, 16170035, 192384876, 2483177808, 34554278857, 515620794591, 8212685046336, 139062777326000, 2494364438359953, 47245095998005059, 942259727190907180, 19737566982241851720, 433234326593362631601, 9943659797649140568863
Offset: 2

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Author

N. J. A. Sloane

Keywords

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.
  • 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).

Crossrefs

Cf. A173184.

Formula

Apparently a(n) = A173184(n)-1, partial sums of A000166. - Ralf Stephan, May 23 2004
a(n) = A136123(n,1). Emeric Deutsch and Vladeta Jovovic, Dec 17 2007
Conjecture: a(n) = A177265(n) if n even, = A177265(n)-1 if n odd. - R. J. Mathar, Jun 27 2022
Conjecture: D-finite with recurrence a(n) -n*a(n-1) +(n-1)*a(n-3)=0. - R. J. Mathar, Jul 01 2022

A010029 Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= floor(n/2)) = number of permutations of 1..n with exactly floor(n/2) - k runs of consecutive pairs up.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 12, 11, 11, 56, 53, 3, 87, 321, 309, 53, 693, 2175, 2119, 11, 680, 5934, 17008, 16687, 309, 8064, 55674, 150504, 148329, 53, 5805, 96370, 572650, 1485465, 1468457, 2119, 95575
Offset: 1

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Author

N. J. A. Sloane

Keywords

Examples

			Triangle begins
     1
     1     1
     3     3
     1    12    11
    11    56    53
     3    87   321    309
    53   693  2175   2119
    11   680  5934  17008   16687
   309  8064 55674 150504  148329
    53  5805 96370 572650 1485465 1468457
  2119 95575 ...
  ...

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.1.

Links

Crossrefs

Cf. A000255, A001277, A001278, A001279, A001280, A000142 (row sums), A136123 (rows reversed).

Programs

  • Maple
    A010029 := proc(n,k)
    add( x^i*( ((1-y)*x-1)/((1-y)*x^2-1) )^i*i!,i=0..n+1) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,floor(n/2)-k) ;
end proc:
seq(seq( A010029(n,k),k=0..floor(n/2)),n=1..12) ; # R. J. Mathar, Jul 01 2022
  • Mathematica
    max = 16; coes = CoefficientList[ Series[ Sum[ n!*(((1 - y)*x^2 - x)/((1 - y)*x^2 - 1))^n, {n, 0, max}], {x, 0, max}, {y, 0, max}], {x, y}]; Table[ Table[ coes[[n, k]] , {k, 1, Floor[(n + 1)/2]}] // Reverse, {n, 2, max - 4}] // Flatten (* Jean-François Alcover, Jan 10 2013, after Vladeta Jovovic *)

Formula

G.f.: Sum_{n>=0} n! *( ((1-y)*x^2-x)/((1-y)*x^2-1) )^n, for the triangle read right-to-left. - Vladeta Jovovic, Nov 21 2007
T(n,k) = A136123(n,[n/2]-k). - R. J. Mathar, Jul 01 2022

A001278 Number of permutations of length n by rises.

Original entry on oeis.org

1, 11, 87, 693, 5934, 55674, 572650, 6429470, 78366855, 1031378445, 14583751161, 220562730171, 3553474061452, 60765835154948, 1099353888345924, 20980355229808524, 421242574828254525, 8876636475162819615, 195887449298481357835, 4517865858233007694865, 108699311713253202373146, 2723633081926998772488606
Offset: 4

Views

Author

N. J. A. Sloane

Keywords

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.
  • 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).

Formula

a(n) = A136123(n,2). Emeric Deutsch and Vladeta Jovovic, Dec 17 2007
Conjecture: D-finite with recurrence a(n) +(-n-3)*a(n-1) +3*(n-2)*a(n-2) +2*(n-1)*a(n-3) +(-6*n+17)*a(n-4) +(-n+5)*a(n-5) +3*(n-4)*a(n-6)=0. - R. J. Mathar, Jul 01 2022

Extensions

More terms from Vladeta Jovovic, Nov 21 2007

A001279 Number of permutations of length n by rises.

Original entry on oeis.org

3, 53, 680, 8064, 96370, 1200070, 15778800, 220047400, 3257228485, 51125192475, 849388162448, 14905775547488, 275697902983860, 5362979000259804, 109488815508733440, 2341353038132316240, 52346701837709016375, 1221458048752142672625, 29697803502485749344120, 751211166036942984639200
Offset: 6

Views

Author

N. J. A. Sloane

Keywords

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.
  • 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).

Formula

a(n) = A136123(n,3). - Emeric Deutsch and Vladeta Jovovic, Dec 17 2007
Conjecture D-finite with recurrence 3*(-n+6)*a(n) +(3*n^2-23*n+67)*a(n-1) +(5*n^2-58*n+179)*a(n-2) +(n^2-51*n+128)*a(n-3) +(-17*n^2+183*n-364)*a(n-4) -4*(n-4)*(n-11)*a(n-5) +12*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, Jul 01 2022

Extensions

More terms from Vladeta Jovovic, Nov 21 2007

A001280 Number of permutations of length n by rises.

Original entry on oeis.org

11, 309, 5805, 95575, 1516785, 24206055, 396475975, 6733084365, 119143997490, 2201649739310, 42514526708766, 857750898213594, 18068801884373310, 397038791150060850, 9090755207499817170, 216635190303090215910
Offset: 8

Views

Author

N. J. A. Sloane

Keywords

References

  • F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264.
  • 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).

Formula

a(n) = A136123(n,4). - Emeric Deutsch and Vladet Jovovic, Dec 17 2007
Conjecture: D-finite with recurrence (n-8)*(n-9)*a(n) -(n-9)*(n^2-6*n-2)*a(n-1) +(2*n^3-33*n^2+143*n-118)*a(n-2) +(5*n^2-27*n+16)*a(n-3) -2*(n-5)*(n-3)*(n-4)*a(n-4) +(n-5)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 01 2022

Extensions

More terms from Vladeta Jovovic, Nov 21 2007
Showing 1-5 of 5 results.

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

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