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.

A182926 Row sums of absolute values of A182928.

Original entry on oeis.org

1, 2, 3, 10, 25, 161, 721, 5706, 40881, 385687, 3628801, 41268613, 479001601, 6324319717, 87212177053, 1317906346186, 20922789888001, 357099708702023, 6402373705728001, 121882752536893635, 2432928081076384321, 51140835669924352717
Offset: 1

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Author

Peter Luschny, Apr 16 2011

Keywords

Comments

The sum of multinomial coefficients can be computed recursively as
A005651(0) = 1 and A005651(n) = Sum_{1<=k<=n} binomial(n-1,k-1) * A182926(k) * A005651(n-k).
Möbius inversion yields: 1, 1, 2, 8, 24, 157, 720, 5696, 40878,...
A182927(2*i+1) = A182926(2*i+1).

Examples

			a(6) = 1 + 10 + 30 + 120 = 161.
		

Crossrefs

Programs

  • Maple
    A182926 := proc(n) local d;
    add(n!/(d*((n/d)!)^d),d = numtheory[divisors](n)) end:
    seq(A182926(i), i = 1..22);
  • Mathematica
    a[n_] := Sum[ Abs[ -n!/(d*(-(n/d)!)^d)], {d, Divisors[n]}]; Table[ a[n], {n, 1, 22}] (* Jean-François Alcover, Jul 29 2013 *)

Formula

a(n) = Sum_{d|n} n!/(d*((n/d)!)^d).
E.g.f.: Sum_{k>=1} log(1/(1 - x^k/k!)). - Ilya Gutkovskiy, May 21 2019

A182927 Row sums of A182928.

Original entry on oeis.org

1, 0, 3, -8, 25, -99, 721, -5704, 40881, -340325, 3628801, -41245511, 479001601, -6129725315, 87212177053, -1317906346184, 20922789888001, -354320889234597, 6402373705728001, -121882630320799633, 2432928081076384321, -51041048673495232715
Offset: 1

Views

Author

Peter Luschny, Apr 16 2011

Keywords

Comments

The number of partitions of an n-set with distinct block sizes can
be computed recursively as A007837(0) = 1 and A007837(n) = - Sum_{1<=k<=n} binomial(n-1,k-1) * A182927(k) * A007837(n-k).
Möbius inversion yields: 1, -1, 2, -8, 24, -101, 720, -5696, 40878,...
A182927(2*i+1) = A182926(2*i+1)

Examples

			a(6) = 1 - 10 + 30 - 120 = -99.
		

Crossrefs

Programs

  • Maple
    A182927 := proc(n) local d;
    add(-n! / (d*(-(n/d)!)^d), d = numtheory[divisors](n)) end:
    seq(A182927(i), i = 1..22);
  • Mathematica
    a[n_] := Sum[ -n!/(d*(-(n/d)!)^d), {d, Divisors[n]}]; Table[a[n], {n, 1, 22}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Formula

a(n) = Sum_{d|n} -n!/(d*(-(n/d)!)^d).
E.g.f.: Sum_{k>=1} log(1 + x^k/k!). - Ilya Gutkovskiy, May 21 2019

A076901 E.g.f.: 1/Product_{m>0} (1+(-x)^m/m!).

Original entry on oeis.org

1, 1, 1, 4, 21, 96, 520, 3795, 32053, 284368, 2763876, 30648465, 373339824, 4833294389, 67167087793, 1009753574739, 16215467043493, 275361718915824, 4947532173402532, 94054153646919213, 1882793796608183356, 39528099512321898363, 869222284280777733043
Offset: 0

Views

Author

Vladeta Jovovic, Nov 26 2002

Keywords

Crossrefs

Programs

  • Maple
    A076901 := proc(n) local a, s;
    s := proc(n) local d; add((-1)^d*n!/(d*(n/d)!^d),
    d = numtheory[divisors](n)) end:
    a := proc(n) option remember; local k;
    `if`(n=0, 1, add(binomial(n-1,k-1)*s(k)*a(n-k),k = 1..n)) end:
    (-1)^n*a(n) end:
    seq(A076901(n), n=0..20);  # Peter Luschny, Apr 16 2011
  • Mathematica
    nmax = 25; Table[SeriesCoefficient[Product[1/(1 + (-x)^k/k!), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 14 2017 *)

Formula

a(n) ~ c * n!, where c = Product_{k>=2} (1 + (-1)^k/k!) = 0.773515873861910082... - Vaclav Kotesovec, Sep 14 2017
Showing 1-3 of 3 results.