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.

A145886 Number of excedances in all odd permutations of {1,2,...,n} with no fixed points.

Original entry on oeis.org

0, 1, 0, 12, 50, 405, 3234, 29680, 300348, 3337425, 40382540, 528644556, 7445076990, 112248853717, 1803999433950, 30788257007040, 556112892188504, 10598857474652865, 212565974908314168, 4475073155964510700
Offset: 1

Views

Author

Emeric Deutsch, Nov 06 2008

Keywords

Examples

			a(4)=12 because the odd derangements of {1,2,3,4} are 4123, 3142, 4312, 2413, 2341 and 3421, having 1, 2, 2, 2, 3 and 2, excedances, respectively.

Links

Crossrefs

Cf. A145880, A145881, A145887.

Programs

  • Maple
    G:=(1/4)*z^2*(2-2*z+z^2)*exp(-z)/(1-z)^2: Gser:=series(G,z=0,30): seq(factorial(n)*coeff(Gser,z,n),n=1..21);
  • Mathematica
    Rest[CoefficientList[Series[1/4*x^2*(2-2*x+x^2)*E^(-x)/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 07 2013 *)

Formula

E.g.f.: (1/4)*z^2*(2-2*z+z^2)*exp(-z)/(1-z)^2.
a(n) = Sum_{k=1..n-1} k * A145880(n,k), n>=2.
a(n) ~ n!*exp(-1)*n/4. - Vaclav Kotesovec, Oct 07 2013
D-finite with recurrence +(-3*n+7)*a(n) +(3*n+2)*(n-3)*a(n-1) +(3*n^2-n+16)*a(n-2) +(3*n^2-23*n+32)*a(n-3) +(3*n-5)*(n-3)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

A145887 Number of excedances in all even permutations of {1,2,...,n} with no fixed points.

Original entry on oeis.org

0, 0, 3, 6, 60, 390, 3255, 29652, 300384, 3337380, 40382595, 528644490, 7445077068, 112248853626, 1803999434055, 30788257006920, 556112892188640, 10598857474652712, 212565974908314339, 4475073155964510510
Offset: 1

Views

Author

Emeric Deutsch, Nov 07 2008

Keywords

Examples

			a(4)=6 because the even derangements of {1,2,3,4} are 3412, 2143 and 4321, having 2, 2 and 2, excedances, respectively.

Links

Crossrefs

Cf. A145880, A145881, A145886.

Programs

  • Maple
    G:=(1/4)*z^3*(2-z)*exp(-z)/(1-z)^2: Gser:=series(G,z=0,30): seq(factorial(n)*coeff(Gser,z,n),n=1..21);
  • Mathematica
    Table[1/4*n*n!*Sum[(-1)^k*(k+2)*(k-1)/(k+1)!, {k,2,n-1}],{n,1,20}] (* Vaclav Kotesovec, Oct 28 2012 *)

Formula

a(n) = Sum_{k=1..n-1} k*A145881(n,k), for n>=2.
E.g.f.: (1/4)*z^3*(2-z)*exp(-z)/(1-z)^2.
a(n) = 1/4*n*n!*Sum_{k=2..n-1} (-1)^k*(k+2)*(k-1)/(k+1)!. - Vaclav Kotesovec, Oct 28 2012
a(n) ~ n * n! / (4*exp(1)). - Vaclav Kotesovec, Dec 10 2021
D-finite with recurrence (-n+3)*a(n) +(n^2-3*n-2)*a(n-1) +(n-1)*(n+1)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

A145880 Triangle read by rows: T(n,k) is the number of odd permutations of {1,2,...,n} with no fixed points and having k excedances (n>=1; k>=1).

Original entry on oeis.org

0, 1, 0, 0, 1, 4, 1, 0, 10, 10, 0, 1, 26, 81, 26, 1, 0, 56, 406, 406, 56, 0, 1, 120, 1681, 3816, 1681, 120, 1, 0, 246, 6210, 26916, 26916, 6210, 246, 0, 1, 502, 21433, 160054, 303505, 160054, 21433, 502, 1, 0, 1012, 70774, 852346, 2747008, 2747008, 852346, 70774
Offset: 1

Views

Author

Emeric Deutsch, Nov 06 2008

Keywords

Comments

Row n has n-1 entries (n>=2).
Sum of entries in row n = A000387(n).
Sum_{k=1..n-1} k*T(n,k) = A145886(n) (n>=2).

Examples

			T(4,2)=4 because the odd derangements of {1,2,3,4} with 2 excedances are 3142, 4312, 2413 and 3421.
Triangle starts:
  0;
  1;
  0,  0;
  1,  4,  1;
  0, 10, 10,  0;
  1, 26, 81, 26,  1;

Links

Crossrefs

Cf. A000387, A145886, A145881, A145887.

Programs

  • Maple
    G:=((1-t)*exp(-t*z)/(1-t*exp((1-t)*z))+(t*exp(-z)-exp(-t*z))/(1-t))*1/2: Gser:=simplify(series(G,z=0,15)): for n to 11 do P[n]:=sort(expand(factorial(n)*coeff(Gser,z,n))) end do: 0; for n to 11 do seq(coeff(P[n],t,j),j=1..n-1) end do; # yields sequence in triangular form

Formula

E.g.f.: ((1-t)*exp(-tz)/(1-t*exp((1-t)z)) + (t*exp(-z)-exp(-tz))/(1-t))/2.

Extensions

Formula corrected by N. J. A. Sloane, Jul 20 2017 at the request of the author.
Showing 1-3 of 3 results.

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

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