A273001 -id:A273001 - cp's OEIS frontend

A205801 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) / (n^2) ).

1, 1, 1, 1, 7, 31, 91, 211, 1681, 52417, 461161, 2427481, 10744471, 219643711, 2619643027, 18939628891, 1410692293921, 23943786881281, 263853697605841, 2237281161036337, 53316533506210471, 900164075618402911, 11265158441537890891, 112769404714319769571

Offset: 0

Paul D. Hanna, Jan 31 2012

nonn

Number of permutations of [n] whose cycle lengths are squares. - Alois P. Heinz, May 12 2016

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + 91*x^6/6! +...
where
log(A(x)) = x + x^4/4 + x^9/9 + x^16/16 + x^25/25 + x^36/36 +...

Alois P. Heinz, Table of n, a(n) for n = 0..451
David Harry Richman and Andrew O'Desky, Derangements and the p-adic incomplete gamma function, arXiv:2012.04615 [math.NT], 2020.
Wikipedia, Liouville Function.

Cf. A000290, A193374, A205800, A205802, A273001, A273997, A308397, A317129, A329945, A353184.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(j),
       a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 12 2016

a[n_] := a[n] = If[n==0, 1, Sum[If[IntegerQ @ Sqrt[j], a[n-j]*(j-1)! * Binomial[n-1, j-1], 0], {j, 1, n}]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 19 2017, after Alois P. Heinz *)
nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^(LiouvilleLambda[k]/k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 17 2019 *)

{a(n)=n!*polcoeff(exp(sum(m=1, sqrtint(n+1), x^(m^2)/(m^2)+x*O(x^n))), n)}

a(n) = if(n==0, 1, (n-1)!*sum(k=1, sqrtint(n), a(n-k^2)/(n-k^2)!)); \\ Seiichi Manyama, Apr 29 2022

The e.g.f. A(x)=1+a(1)x+a(2)x^2/2!+... is equal to the power series expansion of the product of (1-x^n)^{-lambda(n)/n} (n=1,2,...) where lambda(n) is the Liouville function A008836 (follows easily from the Lambert series of lambda(n) - see e. g., the Wikipedia link). - Mamuka Jibladze, Jan 12 2014

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor(sqrt(n))} a(n-k^2)/(n-k^2)!. - Seiichi Manyama, Apr 29 2022

A218002 E.g.f.: exp( Sum_{n>=1} x^prime(n) / prime(n) ).

Original entry on oeis.org

1, 0, 1, 2, 3, 44, 55, 1434, 3913, 39752, 392481, 5109290, 34683451, 914698212, 5777487703, 91494090674, 1504751645265, 31764834185744, 379862450767873, 12634073744624082, 132945783064464691, 2753044719709341980, 64135578414076991031, 1822831113987975441482

Offset: 0

Paul D. Hanna, Oct 17 2012

nonn

Conjecture: a(n) = number of degree-n permutations of prime order.

The conjecture is false. Cf. A214003. This sequence gives the number of n-permutations whose cycle lengths are restricted to the prime numbers. - Geoffrey Critzer, Nov 08 2015

			E.g.f.: A(x) = 1 + x^2/2! + 2*x^3/3! + 3*x^4/4! + 44*x^5/5! + 55*x^6/6! + 1434*x^7/7! + ...
where
log(A(x)) = x^2/2 + x^3/3 + x^5/5 + x^7/7 + x^11/11 + x^13/13 + x^17/17 + x^19/19 + x^23/23 + x^29/29 + ... + x^prime(n)/prime(n) + ...
a(5) = 44 because there are 5!/5 = 24 permutations that are 5-cycles and there are 5!/(2*3) = 20 permutations that are the disjoint product of a 2-cycle and a 3-cycle. - _Geoffrey Critzer_, Nov 08 2015

Alois P. Heinz, Table of n, a(n) for n = 0..450
Ljuben Mutafchiev, A Note on the Number of Permutations whose Cycle Lengths Are Prime Numbers, arXiv:2108.05291 [math.CO], 2021.

Cf. A000040, A214003, A273001, A273998, A317131, A329944.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(isprime(j),
      a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 12 2016

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n], Apply[And, PrimeQ[#]] &]]], {n, 0,23}] (* Geoffrey Critzer, Nov 08 2015 *)

{a(n)=n!*polcoeff(exp(sum(k=1,n,x^prime(k)/prime(k))+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

A193374 E.g.f.: A(x) = exp( Sum_{n>=1} x^(n(n+1)/2) / (n(n+1)/2) ).

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 201, 1191, 4593, 36009, 620721, 5297931, 40360761, 474989373, 4345942329, 122776895151, 2118941145441, 21344580276561, 303071564084193, 4476037678611219, 59935820004483561, 3838519441659950181, 78361805638079449641, 949279542954821272503

Offset: 0

Paul D. Hanna, Jul 24 2011

nonn

Number of permutations of [n] whose cycle lengths are triangular numbers. - Alois P. Heinz, May 12 2016

			E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 21*x^5/5! + 201*x^6/6! +...
where
log(A(x)) = x + x^3/3 + x^6/6 + x^10/10 + x^15/15 + x^21/21 +...

Alois P. Heinz, Table of n, a(n) for n = 0..451

Cf. A000217, A193375, A273001, A305824, A317130.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(issqr(8*j+1),
       a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 12 2016

a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ @ Sqrt[8*j + 1], a[n - j]*(j - 1)!*Binomial[n - 1, j - 1], 0], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)

{a(n)=n!*polcoeff(exp(sum(m=1,sqrtint(2*n+1),x^(m*(m+1)/2)/(m*(m+1)/2)+x*O(x^n))),n)}

A317128 Number of permutations of [n] whose lengths of increasing runs are Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 6, 23, 112, 652, 4425, 34358, 299971, 2910304, 31059715, 361603228, 4560742758, 61947243329, 901511878198, 13994262184718, 230811430415207, 4030772161073249, 74301962970014978, 1441745847111969415, 29374226224980834077, 626971133730275593916

Offset: 0

Alois P. Heinz, Jul 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000045, A097597, A273001, A317111, A317129, A317130, A317131, A317132, A317444.

g:= n-> (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2):
b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
      `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
       add(b(u+j-1, o-j, t+1), j=1..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..27);

g[n_] := With[{t = 5n^2}, If[IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4], 1, 0]];
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, g[t],
     If[g[t] == 1, Sum[b[u - j, o + j - 1, 1], {j, 1, u}], 0] +
     Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 27] (* Jean-François Alcover, Mar 29 2021, after Alois P. Heinz *)

A273994 Number of endofunctions on [n] whose cycle lengths are Fibonacci numbers.

Original entry on oeis.org

1, 1, 4, 27, 250, 2975, 43296, 744913, 14797036, 333393345, 8403026320, 234300271811, 7161316358616, 238108166195263, 8556626831402560, 330494399041444425, 13654219915946513296, 600870384794864432897, 28060233470995898505024, 1386000542545570348128235

Offset: 0

Alois P. Heinz, Jun 06 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000045, A060435, A116956, A273001, A273996, A273997, A273998, A305824.

b:= proc(n) option remember; local r, f, g;
      if n=0 then 1 else r, f, g:= $0..2;
      while f<=n do r:= r+(f-1)!*b(n-f)*
         binomial(n-1, f-1); f, g:= g, f+g
      od; r fi
    end:
a:= n-> add(b(j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);

b[n_] := b[n] = Module[{r, f, g}, If[n == 0, 1, {r, f, g} = {0, 1, 2}; While[f <= n, r = r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1]; {f, g} = {g, f + g}]; r]];
a[0] = 1; a[n_] := Sum[b[j]*n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 06 2018, from Maple *)

A272603 Number of permutations of [n] whose cycle lengths are factorials.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 196, 1072, 7484, 42940, 261496, 1477136, 15219832, 134828344, 1488515120, 13692017536, 130252442896, 1123580329232, 14639510308384, 173489066401600, 2528654220104096, 31472160333513376, 402634734214583872, 4645625988351336704, 25925035549644280991680

Offset: 0

Joerg Arndt, May 29 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..455

Cf. A000142, A273001 (cycle lengths are Fibonacci numbers), A272602 (e.g.f.: exp( sum(n>=1, x^(n!) / n ) ) ), A273996, A317132.

a:= proc(n) option remember; local r, f, i;
      if n=0 then 1 else r, f, i:= $0..2;
        while f<=n do r:= r +a(n-f)*(f-1)!*
          binomial(n-1, f-1); f, i:= f*i, i+1
        od; r
      fi
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 04 2016

nmax = 4; egf = Exp[Sum[x^n!/n!, {n, 1, nmax}]] + O[x]^(nmax! + 1); CoefficientList[egf, x]*Range[0, nmax!]! (* Jean-François Alcover, Feb 19 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(n=1,10,x^(n!)/n!))))

E.g.f.: exp( sum(n>=1, x^(n!) / n! ) ).

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A205801 Expansion of e.g.f. exp( Sum_{n>=1} x^(n^2) / (n^2) ).

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A218002 E.g.f.: exp( Sum_{n>=1} x^prime(n) / prime(n) ).

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A193374 E.g.f.: A(x) = exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2) ).

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A317128 Number of permutations of [n] whose lengths of increasing runs are Fibonacci numbers.

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A273994 Number of endofunctions on [n] whose cycle lengths are Fibonacci numbers.

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Maple

Mathematica

A272603 Number of permutations of [n] whose cycle lengths are factorials.

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A193374 E.g.f.: A(x) = exp( Sum_{n>=1} x^(n(n+1)/2) / (n(n+1)/2) ).