cp's OEIS Frontend

A293588 E.g.f.: exp(x + x^6/6).

%I A293588 #18 Mar 08 2021 02:56:38
%S A293588 1,1,1,1,1,1,121,841,3361,10081,25201,55441,6763681,86692321,
%T A293588 605765161,3027624601,12109056961,41169011521,5063607974881,
%U A293588 94197184734241,939457659787201,6572292677455681,36141156689382361,166238526616664041,20612479896229156321
%N A293588 E.g.f.: exp(x + x^6/6).
%C A293588 These are the telephone numbers T^(6)_n of [Artioli et al., p. 7].
%H A293588 Seiichi Manyama, <a href="/A293588/b293588.txt">Table of n, a(n) for n = 0..524</a>
%H A293588 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.
%F A293588 a(n) = a(n-1) + (n-1)!/(n-6)! * a(n-6).
%F A293588 a(n) = Sum_{j=0..floor(n/6)} n!/(6^j*j!*(n-6*j)!). - _G. C. Greubel_, Mar 07 2021
%t A293588 With[{nn=30},CoefficientList[Series[Exp[x+x^6/6],{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Dec 11 2017 *)
%t A293588 Table[Sum[n!/(6^k*k!*(n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* _G. C. Greubel_, Mar 07 2021 *)
%o A293588 (PARI) my(x = 'x + O('x^30)); Vec(serlaplace(exp(x + x^6/6))) \\ _Michel Marcus_, Oct 13 2017
%o A293588 (Sage)
%o A293588 f=factorial;
%o A293588 [sum( f(n)/(6^k*f(k)*f(n-6*k)) for k in [0..n/3]) for n in [0..30]] # _G. C. Greubel_, Mar 07 2021
%o A293588 (Magma)
%o A293588 F:= Factorial;
%o A293588 [(&+[ F(n)/(6^k*F(k)*F(n-6*k)): k in [0..Floor(n/3)]]): n in [0..30]]; // _G. C. Greubel_, Mar 07 2021
%Y A293588 Sequences with e.g.f. exp(x + x^m/m): A000079 (m=1), A000085 (m=2), A001470 (m=3), A118934 (m=4), A052501 (m=5), this sequence (m=6), A053497 (m=7).
%Y A293588 Cf. A000085, A001470, A052501, A118934.
%K A293588 nonn
%O A293588 0,7
%A A293588 _Eric M. Schmidt_, Oct 12 2017