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%I #77 Feb 23 2021 18:03:45
%S 1,1,5,31,241,2261,24781,309835,4342241,67308841,1141960501,
%T 21026890391,417264626065,8871853115581,201100863674621,
%U 4838817223845571,123128720142540481,3302478863343928145,93091427773284348901,2750635764338982054031,84994418675445218025521
%N Expansion of e.g.f.: exp( x/(1-x)^2 ).
%C Old name: A binomial sum.
%C 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
%C 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
%C The above conjecture is true - see the Bala link. - _Peter Bala_, Jan 20 2018
%H Seiichi Manyama, <a href="/A082579/b082579.txt">Table of n, a(n) for n = 0..433</a>
%H P. Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>
%F a(n) = n!*Sum_{k=0..n} binomial(n+k-1, 2*k-1)/k!.
%F 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.
%F E.g.f.: exp( x/( 1 - x )^2 ).
%F 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
%F 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
%F 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
%F E.g.f.: exp(Sum_{k>=1} k*x^k). - _Vaclav Kotesovec_, Mar 07 2015
%F 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
%F 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
%F a(n) = n*n!*Hypergeometric2F2([1-n, n+1], [3/2, 2], -1/4) with a(0) = 1. - _G. C. Greubel_, Feb 23 2021
%t nn=20;Range[0,nn]!CoefficientList[Series[Exp[ x/(1-x)^2],{x,0,nn}],x] (* _Geoffrey Critzer_, Nov 04 2012 *)
%t nn = 20; Range[0, nn]! * CoefficientList[Series[Product[Exp[k*x^k], {k, 1, nn}], {x, 0, nn}], x] (* _Vaclav Kotesovec_, Mar 21 2016 *)
%t 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 *)
%o (Maxima)
%o a(n):=n!*sum(binomial(n+k-1,2*k-1)/k!,k,1,n); \\ _Vladimir Kruchinin_, Apr 21 2011
%o (PARI)
%o my(x='x+O('x^33));
%o Vec(serlaplace(exp( x/(1-x)^2 )))
%o /* _Joerg Arndt_, Sep 14 2012 */
%o (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
%o (Magma)
%o A082579:= func< n | n eq 0 select 1 else (&+[Factorial(n)*Binomial(n+k-1, n-k)/Factorial(k): k in [1..n]]) >;
%o [A082579(n): n in [0..25]]; // _G. C. Greubel_, Feb 23 2021
%Y Cf. A000262, A052897, A255806, A255807, A255819.
%K easy,nonn
%O 0,3
%A _Emanuele Munarini_, May 07 2003
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