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%I #28 Mar 08 2021 02:56:18
%S 1,1,1,1,7,31,91,211,1681,12097,57961,209881,1874071,17842111,
%T 117303187,575683291,5691897121,65641390081,544238393041,
%U 3362783785777,36455473647271,485442581801311,4828464958268491,35900587138847971
%N E.g.f.: exp(x + x^4/4).
%C Equals row sums of triangle A118933.
%C These are the telephone numbers T^(4)_n of [Artioli et al., p. 7]. - _Eric M. Schmidt_, Oct 12 2017
%H Vincenzo Librandi, <a href="/A118934/b118934.txt">Table of n, a(n) for n = 0..200</a>
%H Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, <a href="https://arxiv.org/abs/1703.07262">Motzkin Numbers: an Operational Point of View</a>, arXiv:1703.07262 [math.CO], 2017.
%H Vaclav Kotesovec, <a href="/A118934/a118934.jpg">Graph - asymptotic (20000 terms)</a>
%F a(n) = a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4) for n>=4, with a(0)=a(1)=a(2)=a(3)=1.
%F a(n) ~ 1/2 * n^(3*n/4) * exp(n^(1/4)-3*n/4). - _Vaclav Kotesovec_, Feb 25 2014
%F a(n) = Sum_{k=0..floor(n/4)} n!/(4^k*k!*(n-4*k)!). - _G. C. Greubel_, Mar 07 2021
%t With[{nn=30},CoefficientList[Series[Exp[x+x^4/4],{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Jan 26 2013 *)
%t Table[Sum[n!/(4^k*k!*(n-4*k)!), {k,0,n/4}], {n,0,30}]
%o (PARI) a(n)=if(n<0,0,if(n==0,1,a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4)))
%o (Sage) f=factorial; [sum(f(n)/(4^j*f(j)*f(n-4*j)) for j in (0..n/4)) for n in (0..30)] # _G. C. Greubel_, Mar 07 2021
%o (Magma) F:=Factorial; [(&+[F(n)/(4^j*F(j)*F(n-4*j)): j in [0..Floor(n/4)]]): n in [0..30]]; // _G. C. Greubel_, Mar 07 2021
%Y Sequences with e.g.f. exp(x + x^m/m): A000079 (m=1), A000085 (m=2), A001470 (m=3), this sequence (m=4), A052501 (m=5), A293588 (m=6), A053497 (m=7).
%Y Cf. A118933.
%K nonn
%O 0,5
%A _Paul D. Hanna_, May 06 2006
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