A082579 Expansion of e.g.f.: exp( x/(1-x)^2 ).

1, 1, 5, 31, 241, 2261, 24781, 309835, 4342241, 67308841, 1141960501, 21026890391, 417264626065, 8871853115581, 201100863674621, 4838817223845571, 123128720142540481, 3302478863343928145, 93091427773284348901, 2750635764338982054031, 84994418675445218025521

Offset: 0

Author

Emanuele Munarini, May 07 2003

Keywords

• easy
• nonn

Comments

Old name: A binomial sum.

a(n) is the number of ways that n people can form any number of lines and then designate one person in each line. Equivalently, number of ways to linearly arrange the elements in each block of a set partition, then underline one element in each block summed over all set partitions of {1,2,...,n}. a(2) = 5: [1'][2'], [1',2], [1,2'], [2',1], [2,1']. - Geoffrey Critzer, Nov 04 2012

It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with period dividing k. - Peter Bala, Nov 14 2017

The above conjecture is true - see the Bala link. - Peter Bala, Jan 20 2018

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..433
• Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

Crossrefs

Cf. A000262, A052897, A255806, A255807, A255819.

Programs

Magma

A082579:= func< n | n eq 0 select 1 else (&+[Factorial(n)*Binomial(n+k-1, n-k)/Factorial(k): k in [1..n]]) >;
[A082579(n): n in [0..25]]; // G. C. Greubel, Feb 23 2021

Mathematica

nn=20;Range[0,nn]!CoefficientList[Series[Exp[ x/(1-x)^2],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 04 2012 *)
nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k*x^k], {k, 1, nn}], {x, 0, nn}], x] (* Vaclav Kotesovec, Mar 21 2016 *)
Table[If[n==0, 1, n*n!*HypergeometricPFQ[{1-n, n+1}, {3/2, 2}, -1/4]], {n, 0, 25}] (* G. C. Greubel, Feb 23 2021 *)

Maxima

a(n):=n!*sum(binomial(n+k-1,2*k-1)/k!,k,1,n); /* Vladimir Kruchinin, Apr 21 2011 */

PARI

my(x='x+O('x^33));
Vec(serlaplace(exp( x/(1-x)^2 )))
/* Joerg Arndt, Sep 14 2012 */

Sage

[1 if n==0 else factorial(n)*sum( binomial(n+k-1, n-k)/factorial(k) for k in (1..n)) for n in (0..25)] # G. C. Greubel, Feb 23 2021

Formula

a(n) = n!*Sum_{k=0..n} binomial(n+k-1, 2*k-1)/k!.

Recurrence: a(n+3) - (3*n+7)*a(n+2) + (n+2)*(3*n+2)*a(n+1) - (n+2)*(n+1)*n*a(n) = 0.

E.g.f.: exp( x/( 1 - x )^2 ).

Special values of the hypergeometric function 2F2: a(n)=n!*n*hypergeom([n+1, -n+1], [3/2, 2], -1/4), n >= 1. - Karol A. Penson, Jan 29 2004

a(n) ~ 2^(1/6)*n^(n-1/6)*exp(-1/12 + 3*(n/2)^(2/3) - n)/sqrt(3). - Vaclav Kotesovec, Jun 26 2013

E.g.f.: E(0)/2, where E(k) = 1 + 1/( 1 - x/(x + (1-x)^2*(k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013

E.g.f.: exp(Sum_{k>=1} k*x^k). - Vaclav Kotesovec, Mar 07 2015

a(n) = n!*y(n), with y(0) = 1, y(n) = (Sum_{k=0..n-1} (n-k)^2*y(k))/n. - Benedict W. J. Irwin, Jun 02 2016

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(J_2(k)/k), where J_2() is the Jordan function (A007434). - Ilya Gutkovskiy, May 25 2019

a(n) = n*n!*Hypergeometric2F2([1-n, n+1], [3/2, 2], -1/4) with a(0) = 1. - G. C. Greubel, Feb 23 2021