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.

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A102289 Total number of odd lists in all sets of lists, cf. A000262.

Original entry on oeis.org

0, 1, 2, 15, 76, 665, 5286, 56287, 597080, 7601841, 99702730, 1484554511, 23049638052, 393702612745, 7036703742446, 135702811542495, 2737989749177776, 58848546456947297, 1321063959370833810, 31310238786268648591, 773291778432688011260, 20031956775840631151481
Offset: 0

Views

Author

Vladeta Jovovic, Feb 19 2005

Keywords

Links

Crossrefs

Cf. A052852, A066897, A066898, A059570, A059570, A081358, A092691.

Programs

  • Maple
    G:=(x/(1-x^2))*exp(x/(1-x)): Gser:=series(G,x=0,25): seq(n!*coeff(Gser,x^n),n=1..22); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::odd, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016
  • Mathematica
    Rest[CoefficientList[Series[x/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
nxt[{n_,a_,b_,c_}]:={n+1,b,c,(n+1)*c+(n+1)^2*b-(n-1)^2 (n+1)*a}; NestList[ nxt,{2,0,1,2},30][[All,2]] (* Harvey P. Dale, Jan 13 2019 *)

Formula

E.g.f.: x/(1-x^2)*exp(x/(1-x)).
a(n) = n*a(n-1) + n^2*a(n-2) - (n-2)^2*n*a(n-3). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 + 7/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013

Extensions

More terms from Emeric Deutsch, Jun 24 2005
a(0)=0 pepended by Alois P. Heinz, May 10 2016

A102290 Total number of even lists in all sets of lists, cf. A000262.

Original entry on oeis.org

0, 0, 2, 6, 60, 380, 3990, 37002, 450296, 5373720, 76018410, 1096730030, 17814654132, 299645294676, 5511836578430, 105550556136690, 2171244984679920, 46545825736022192, 1059273836225051346, 25100215228045842390, 626204775725372971820, 16239127347086448236460
Offset: 0

Views

Author

Vladeta Jovovic, Feb 19 2005

Keywords

Links

Crossrefs

Cf. A052852, A059570, A066897, A066898, A081358, A092691.

Programs

  • Magma
    l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[0,0]cat[Factorial(n)*(&+[(-1)^(n+j)*l(j,-1): j in [0..n-2]]): n in [2..30]]; // G. C. Greubel, Mar 09 2021
  • Maple
    Gser:=series(x^2*exp(x/(1-x))/(1-x^2),x=0,22):seq(n!*coeff(Gser,x^n),n=1..21); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::even, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016
  • Mathematica
    Rest[CoefficientList[Series[x^2/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
Table[If[n<2, 0, n!*Sum[(-1)^(n-j)*LaguerreL[j, -1], {j,0,n-2}]], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)
  • Sage
    [0,0]+[factorial(n)*sum((-1)^(n+j)*gen_laguerre(j,0,-1) for j in (0..n-2)) for n in (2..30)] # G. C. Greubel, Mar 09 2021

Formula

E.g.f.: x^2/(1-x^2)*exp(x/(1-x)).
Recurrence: (n-2)*a(n) = (n-2)*n*a(n-1) + (n-1)^2*n*a(n-2) - (n-3)*(n-2)*(n-1)*n*a(n-3). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 - 41/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
a(n) = n! * Sum_{j=0..n-2} (-1)^(n+j)*LaguerreL(j, -1) for n>1 with a(0)=a(1)=0. - G. C. Greubel, Mar 09 2021

Extensions

More terms from Emeric Deutsch, Mar 27 2005
a(0)=0 prepended by Alois P. Heinz, May 10 2016

A115201 Number of even parts of partitions of n in the Abramowitz-Stegun (A-St) order.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 1, 3, 0, 2, 1, 0, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 3, 0, 2, 1, 0, 1, 0, 2, 0, 2, 1, 1, 1, 3, 1, 0, 2, 0, 2, 4, 1, 1, 3, 0, 2, 1, 0, 0, 1, 1, 1, 1, 0, 2, 0, 2, 2, 2, 0, 1, 1, 1
Offset: 0

Views

Author

Wolfdieter Lang, Feb 23 2006

Keywords

Comments

A conjugacy class of the symmetric group S_n with the cycle structure given by the partition, listed in the A-St order, consists of even, resp. odd, permutations if a(n,m) is even, resp. odd.
See A115198 for the parity of a(n,m) with 1 for even, 0 for odd (main entry).
See A115199 for the parity of a(n,m) with 0 for even, 1 for odd.
The parity of these numbers determines whether a conjugacy class of the symmetric group S_n, which is determined by its cycle structure, consists of even or odd permutations.
The row length sequence of this triangle is p(n)=A000041(n) (number of partitions).

Examples

			[0];[1, 0];[0, 1, 0];[1, 0, 2, 1, 0];[0, 1, 1, 0, 2, 1, 0];...

Links

Crossrefs

The sequence of row lengths is A066898 (total number of even parts in all partitions of n).

Formula

a(n,m) = Sum_{j=1..floor(n/2)} e(n,m,2*j) with the exponents e(n,m,k) of the m-th partition of n in the A-St order; i.e. the sum of the exponents of the even parts of the partition (1^e(n,m,1),2^e(n,m,2),..., n^e(n,m,n)).
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