A193374 -id:A193374 - 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

A317130 Number of permutations of [n] whose lengths of increasing runs are triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 7, 24, 93, 483, 2832, 17515, 123226, 978405, 8312802, 75966887, 756376739, 8070649675, 91320842018, 1099612368110, 14054043139523, 189320856378432, 2682416347625463, 39945105092501742, 623240458310527252, 10160826473676346731, 172871969109661492526

Offset: 0

Alois P. Heinz, Jul 21 2018

nonn

			a(2) = 1: 21.
a(3) = 2: 123, 321.
a(4) = 7: 1243, 1342, 2134, 2341, 3124, 4123, 4321.
a(5) = 24: 12543, 13542, 14532, 21354, 21453, 23541, 24531, 31254, 31452, 32145, 32451, 34521, 41253, 41352, 42135, 42351, 43125, 51243, 51342, 52134, 52341, 53124, 54123, 54321.

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

Cf. A000217, A097597, A193374, A317111, A317128, A317129, A317131, A317132, A317446.

g:= n-> `if`(issqr(8*n+1), 1, 0):
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_] := If[IntegerQ @ Sqrt[8n+1], 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, Apr 29 2020, after Alois P. Heinz *)

A273001 Number of permutations of [n] whose cycle lengths are Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 6, 18, 90, 420, 2220, 19020, 130860, 1096920, 9862920, 83843640, 1411202520, 16144792560, 203091829200, 2989264122000, 37012939750800, 597962683188000, 8681244913692000, 126467701221607200, 5006833609034743200, 95602098255580238400

Offset: 0

Alois P. Heinz, May 12 2016

nonn

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

Cf. A000045, A193374, A205801, A218002, A272603, A273994, A317128.

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(issqr(5*j^2+4) or issqr(5*j^2-4),
       a(n-j)*(j-1)!*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n == 0, 1, Sum[If[IntegerQ @ Sqrt[5*j^2+4] || IntegerQ @ Sqrt[5*j^2-4], a[n-j]*(j-1)!*Binomial[n-1, j-1], 0], {j, 1, n}]]; Table[ a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 30 2017, translated from Maple *)

E.g.f.: exp(Sum_{n>=2} x^F(n)/F(n)) with F = A000045.

A305824 Number of endofunctions on [n] whose cycle lengths are triangular numbers.

Original entry on oeis.org

1, 1, 3, 18, 157, 1776, 24807, 413344, 8004537, 176630400, 4374300331, 120136735104, 3623854678677, 119102912981248, 4236492477409935, 162152320065532416, 6645233337842716273, 290321208589666369536, 13469914225467040015827, 661442143465113960448000

Offset: 0

Alois P. Heinz, Jun 10 2018

nonn

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

Cf. A000217, A060435, A116956, A193374, A205799, A273994, A273996, A273998.

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, f, g:= r+(f-1)!*
         b(n-f)*binomial(n-1, f-1), f+g, g+1
      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, f, g} = {r + (f - 1)!*b[n - f]*Binomial[n - 1, f - 1], f + g, g + 1}]; 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 15 2018, after Alois P. Heinz *)

A205799 E.g.f.: exp( Sum_{n>=1} x^(n(n+1)/2) / (n(n+1)/2)! ).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 32, 113, 365, 1373, 6072, 25279, 115633, 606321, 3051413, 16344785, 98402881, 576283953, 3523586227, 23840955908, 158428389359, 1085566420290, 8128568533790, 60203101002122, 455911264482697, 3734114950288571, 30413492882578846

Offset: 0

Paul D. Hanna, Jan 31 2012

nonn

Number of set partitions of [n] whose block lengths are triangular numbers. - Alois P. Heinz, Jun 10 2018

			E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 11*x^5/5! + 32*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..603

Cf. A193374, A205802, A305824.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      issqr(8*j+1), a(n-j)*binomial(n-1, j-1), 0), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 10 2018

m = 30;
CoefficientList[Exp[Sum[x^(n(n+1)/2)/(n(n+1)/2)!, {n, 1, m}]] + O[x]^m, x]* Range[0, m-1]! (* Jean-François Alcover, Mar 05 2021 *)

{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)}

A193375 E.g.f.: A(x) = exp( Sum_{n>=1} x^(n*(n+1)/2) ).

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 1201, 7771, 30577, 514585, 8089921, 63701551, 832599241, 14055894997, 137066892145, 3084240161731, 70859008063201, 849408115312561, 15997591979202817, 358582896987674455, 6017079190150763641, 209473179919282488301

Offset: 0

Paul D. Hanna, Jul 24 2011

nonn

			E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 25*x^4/4! + 61*x^5/5! +...
where
log(A(x)) = x + x^3 + x^6 + x^10 + x^15 + x^21 +...

Seiichi Manyama, Table of n, a(n) for n = 0..448

Cf. A010054, A193374, A205800.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A317130 Number of permutations of [n] whose lengths of increasing runs are triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A273001 Number of permutations of [n] whose cycle lengths are Fibonacci numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A305824 Number of endofunctions on [n] whose cycle lengths are triangular numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A205799 E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / (n*(n+1)/2)! ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A193375 E.g.f.: A(x) = exp( Sum_{n>=1} x^(n*(n+1)/2) ).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A205799 E.g.f.: exp( Sum_{n>=1} x^(n(n+1)/2) / (n(n+1)/2)! ).