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.
a:= func< n | NumberOfPartitions(n)*Factorial(n) >; [ a(n) : n in [0..25]]; // Vincenzo Librandi, Jan 17 2019
seq(count(Permutation(n))*count(Partition(n)),n=1..20); # Zerinvary Lajos, Oct 16 2006 with(combinat): A053529 := proc(n): n! * numbpart(n) end: seq(A053529(n), n=0..20); # Johannes W. Meijer, Jul 28 2016
Table[PartitionsP[n] n!, {n, 0, 20}] (* T. D. Noe, Jun 19 2012 *)
N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, x^k/(1-x^k)/k)))) \\ Joerg Arndt, Apr 16 2010
N=66; x='x+O('x^N); Vec(serlaplace(sum(n=0, N, x^n/prod(k=1,n,1-x^k)))) \\ Joerg Arndt, Jan 29 2011
a(n) = n!*numbpart(n); \\ Michel Marcus, Jul 28 2016
from math import factorial from sympy import npartitions def A053529(n): return factorial(n)*npartitions(n) # Chai Wah Wu, Jul 10 2023
s:= proc(n, i) option remember; `if`(i=0, [[]],
map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
end:
a:= proc(n) (l-> add(add(`if`([true$n]=[seq(evalb(
f[g[f[i]]]=f[f[f[i]]]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
a[n_] := a[n] = If[n == 0, 1, Module[{f, g, T}, T = Tuples[Range[n], n]; Table[f = T[[j, #]] &; g = T[[k, #]] &; Table[True, {n}] == Table[f[g[f[i]]] == f[f[f[i]]], {i, n}], {j, n^n}, {k, n^n}] // Flatten // Count[#, True] &]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 5}] (* Jean-François Alcover, Sep 24 2022 *)
s:= proc(n, i) option remember; `if`(i=0, [[]],
map(x-> seq([j, x[]], j=1..n), s(n, i-1)))
end:
a:= proc(n) (l-> add(add(`if`([true$n]=[seq(evalb(
f[g[f[i]]]=g[g[f[i]]]), i=1..n)], 1, 0), g=l), f=l))(s(n$2))
end:
seq(a(n), n=0..5); # Alois P. Heinz, Jul 16 2014
a[n_] := a[n] = If[n == 0, 1, Module[{f, g, T}, T = Tuples[Range[n], n]; Table[f = T[[j, #]]&; g = T[[k, #]]&; Table[True, {n}] == Table[f[g[f[i]]] == g[g[f[i]]], {i, n}], {j, n^n}, {k, n^n}] // Flatten // Count[#, True]&]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 5}] (* Jean-François Alcover, Sep 22 2022 *)
a:= proc(n) option remember; local b; b:=
proc(m, i) option remember; `if`(m=0, n^i, `if`(i<1, 0,
add(b(m-j, i-1)*binomial(m, j)*j, j=0..m)))
end: forget(b):
b(n$2)
end:
seq(a(n), n=0..20); # Alois P. Heinz, Jul 23 2014
a[n_] := If[n==0, 1, Sum[k! Binomial[n, k] (n k)^(n - k), {k, 1, n}]]
Table[a[n],{n,20}] (* David Einstein, Oct 10 2016 *)
a(0) = a(1) = 1 since there is only one endofunction for n=0 or 1 and the equation is satisfied trivially. For n=2, each endofunction f on {1,2} is represented by [f(1),f(2)]. The list of a(2) = 6 pairs (f,g) which satisfy the equation is ([1,1], [1,1]), ([1,1], [1,2]), ([1,2], [1,2]), ([1,2], [2,1]), ([2,2], [1,2]), ([2,2], [2,2]). - _Michael Somos_, Mar 26 2014
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
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 *)
g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= n-> add(binomial(n, k)*Stirling2(n, k)*k!*
add(binomial(n-k, i)*binomial(k, i)*i!*
g(k-i)*n^(n-k-i), i=0..min(k, n-k)), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Jul 18 2014
g[n_] := g[n] = If[n < 2, 1, g[n-1] + (n-1)*g[n-2]];
a[n_] := If[n == 0, 1, Sum[Binomial[n, k]*StirlingS2[n, k]*k!*Sum[ Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]} ], {k, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)
b(n,f) = {sum(k=0, n\2, f(k)*binomial(n,2*k)*(2*k)!/(k!*2^k))}
a(n) = {b(n, k->b(n-2*k, j->1)*b(k, j->2^(k-j)))}
seq(n)=Vec(serlaplace(exp(x + 3*x^2/2 + x^4/4 + O(x*x^n))))
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