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%I #55 Apr 17 2023 16:30:18
%S 1,1,5,13,73,281,1741,8485,57233,328753,2389141,15539261,120661465,
%T 866545993,7140942173,55667517781,484124048161,4046845186145,
%U 36967280461093,328340133863533,3137853448906601,29405064157989241
%N E.g.f.: exp(x + 2*x^2).
%C Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.
%C Combinatorial interpretation: a(n) counts the partitions of a set of n distinguishable objects into subsets of size 1 and 2 with the additional feature that the constituents of the subset of size 2 acquire 2 colors. - _Karol A. Penson_ and P. Blasiak (blasiak(AT)lptl.jussieu.fr), Jun 03 2006
%C In general, e.g.f. exp(x+m*x^2) has general term sum{k=0..n, C(n,k)*m^k*(n-k)!/(n-m*k)!}. [_Paul Barry_, Nov 07 2008]
%C The sequence terms have the form 4*m + 1 (follows from the recurrence). a(n+k) = a(n) (mod k) holds for all n and k by an induction argument making use of the recurrence equation. For each k the sequence a(n) taken modulo k is thus periodic with exact period dividing k. - _Peter Bala_, Nov 15 2017
%H Seiichi Manyama, <a href="/A115329/b115329.txt">Table of n, a(n) for n = 0..665</a> (terms 0..200 from Vincenzo Librandi)
%H Magdalena Boos, Giovanni Cerulli Irelli, Francesco Esposito, <a href="https://arxiv.org/abs/1802.06425">Parabolic orbits of 2-nilpotent elements for classical groups</a>, arXiv:1802.06425 [math.RT], 2018.
%F Term-by-term square equals A115330 which has e.g.f.: exp(x/(1-4*x))/sqrt(1-16*x^2).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)2^k*n!/(n-k)! = Sum_{k=0..n} C(n,k)2^k*(n-k)!/(n-2k)!. - _Paul Barry_, Nov 07 2008
%F a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+8*x)*d/dx. Cf. A000085 and A047974. - _Peter Bala_, Dec 07 2011
%F Conjecture: a(n)-a(n-1)+4*(1-n)*a(n-2)=0. - _R. J. Mathar_, Dec 10 2011
%F a(n) ~ 2^(n-1/2)*exp(sqrt(n)/2-n/2-1/16)*n^(n/2). - _Vaclav Kotesovec_, Oct 19 2012
%F G.f.: 1/Q(0), where Q(k)= 1 + 4*x*k - x/(1 - 4*x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 17 2013
%F G.f.: 1/G(0), where G(k)= 1 - x - 4*(k+1)*x^2/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 21 2013
%F a(n) = i^(1 - n)*2^(3*(n - 1)/2)*KummerU((1 - n)/2, 3/2, -1/8). - _Peter Luschny_, Nov 21 2017
%p a := n -> I^(1 - n)*2^((3*(n - 1))/2)*KummerU((1 - n)/2, 3/2, -1/8):
%p seq(simplify(a(n)), n=0..21); # _Peter Luschny_, Nov 21 2017
%t Range[0, 20]! CoefficientList[Series[Exp[(x + 2 x^2)], {x, 0, 20}], x] (* _Vincenzo Librandi_, May 22 2013 *)
%o (PARI) a(n)=local(m=4);n!*polcoeff(exp(x+m/2*x^2+x*O(x^n)),n)
%Y Column k=4 of A359762.
%Y Cf. A115330, A000085, A047974.
%K nonn,easy
%O 0,3
%A _Paul D. Hanna_, Jan 20 2006
%E More terms from _Karol A. Penson_ and P. Blasiak (blasiak(AT)lptl.jussieu.fr), Jun 03 2006
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