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.

A239761 Number of pairs of functions (f, g) on a set of n elements into itself satisfying f(g(x)) = f(x).

Original entry on oeis.org

1, 1, 10, 159, 3496, 98345, 3373056, 136535455, 6371523712, 336784920849, 19888195110400, 1297716672601151, 92721494240225280, 7199830049013964921, 603715489091812335616, 54366622743565012989375, 5233114241479255004839936, 536180296483497244155041825
Offset: 0

Views

Author

Chad Brewbaker, Mar 26 2014

Keywords

Links

Crossrefs

Cf. A181162, A239769, A239773.
Column k=1 of A245910.
Cf. A001622, A295188.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)*binomial(n, j)*j^j, j=0..n)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 17 2014
  • Mathematica
    f4[n_] := Sum[n^k Sum[Binomial[n - 1, j]*n^(n - 1 - j)*StirlingS1[j + 1, k] *(-1)^(j + k + 1), {j, 0, n - 1}], {k, 1, n}] (* David Einstein, Oct 31 2016 *)

Formula

a(n) ~ 5^(-1/4) * ((1+sqrt(5))/2)^(3*n-1/2) * n^n / exp(2*n/(1+sqrt(5))). - Vaclav Kotesovec, Aug 07 2014
a(n) = Sum_{k = 1..n} A060281(n,k) n^k. - David Einstein, Oct 31 2016
a(n) = n! * [x^n] 1/(1 + LambertW(-x))^n. - Ilya Gutkovskiy, Oct 03 2017

Extensions

a(6)-a(7) from Giovanni Resta, Mar 28 2014
a(8)-a(17) from Alois P. Heinz, Jul 17 2014

A066399 From reversion of e.g.f. for squares.

Original entry on oeis.org

0, 1, -4, 39, -616, 13505, -379296, 12995983, -525688192, 24519144609, -1295527513600, 76481653648631, -4989249262503936, 356408413864589281, -27670449142629400576, 2319870547729387929375, -208886312501433616531456, 20104397299878424990749377
Offset: 0

Views

Author

N. J. A. Sloane, Dec 25 2001

Keywords

Links

Crossrefs

Cf. A295188.

Programs

  • Maple
    read transforms; add(n^2*x^n/n!,n=1..30); series(%,x,31): seriestoseries(%,'revogf'); SERIESTOLISTMULT(%);
with(powseries):powcreate(t(n)=n^2/n!):seq(n!*coeff(tpsform(reversion(t),x,19),x,n),n=0..18); spec:=[A,{A=Prod(Z,Set(A),Set(B)),B=Cycle(A)},labeled];seq(combstruct[count](spec,size=n), n=0..18); # Vladeta Jovovic, May 29 2007
a := n -> `if`(n<2,n,(-2)^(n-1)*doublefactorial(2*n-3)*hypergeom([1-n],[2-2*n],n)): seq(simplify(a(n)),n=0..18); # Peter Luschny, Oct 16 2015
  • Mathematica
    A066399[0] = 0; A066399[1] = 1; A066399[n_] := (-2)^(n - 1) (2 n - 3)!! Hypergeometric1F1[1 - n, 2 - 2 n, n]; Table[A066399[n], {n, 0, 10}] (* Vladimir Reshetnikov, Oct 16 2015 *)
  • PARI
    a(n) = if(n==0, 0, (-1)^(n-1)*(n-1)! * sum(k=0, n-1, (n)^k/k! * binomial(2*n-2-k,n-1))) \\ Altug Alkan, Oct 16 2015

Formula

a(n+1) = (-1)^n*(n)! * Sum_{m=0..n} (n+1)^m/m! * binomial(2*n-m,n). - Vladimir Kruchinin, Feb 22 2011
For n>=2, a(n) = (-2)^(n-1)*(2n-3)!!*hypergeom([1-n], [2-2n], n), where n!! denotes the double factorial A006882. - Vladimir Reshetnikov, Oct 16 2015
E.g.f. g(x) satisfies (g(x) + g(x)^2)*exp(g(x)) = x. - Robert Israel, Oct 16 2015
a(n) ~ (-1)^(n-1) * (2 + sqrt(5))^(n-1/2) * n^(n-1) / (5^(1/4) * exp((sqrt(5) - 1)*n/2)). - Vaclav Kotesovec, Oct 18 2015

A295183 a(n) = n! * [x^n] exp(n*x)/(1 - x)^n.

Original entry on oeis.org

1, 2, 18, 276, 5960, 165870, 5648832, 227507336, 10577029248, 557457222330, 32843470246400, 2139014862736092, 152592485390272512, 11833139429253625574, 991101777088623943680, 89164680959505831930000, 8575295241502192869343232, 877955050581430468997781234, 95337079570413427211596726272
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 16 2017

Keywords

Comments

The n-th term of the n-fold exponential convolution of A000522 with themselves.

Links

Crossrefs

Cf. A000407, A000522, A001813, A081923, A295182, A295188.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x]/(1 - x)^n, {x, 0, n}], {n, 0, 18}]

Formula

a(n) ~ phi^(3*n + 1/2) * n^n / (5^(1/4) * exp(n/phi)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017
a(n) = (-1)^n*n!*Laguerre(n,-2*n,n). - Ilya Gutkovskiy, May 24 2018
Showing 1-3 of 3 results.

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

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