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

1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 55441, 6763681, 86692321, 605765161, 3027624601, 12109056961, 41169011521, 5063607974881, 94197184734241, 939457659787201, 6572292677455681, 36141156689382361, 166238526616664041, 20612479896229156321

Offset: 0

Author

Eric M. Schmidt, Oct 12 2017

Keywords

• nonn

Comments

These are the telephone numbers T^(6)_n of [Artioli et al., p. 7].

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..524
• Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.

Crossrefs

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).

Cf. A000085, A001470, A052501, A118934.

Programs

Magma

F:= Factorial;
[(&+[ 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

Mathematica

With[{nn=30},CoefficientList[Series[Exp[x+x^6/6],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 11 2017 *)
Table[Sum[n!/(6^k*k!*(n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* G. C. Greubel, Mar 07 2021 *)

PARI

my(x = 'x + O('x^30)); Vec(serlaplace(exp(x + x^6/6))) \\ Michel Marcus, Oct 13 2017

Sage

f=factorial;
[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

Formula

a(n) = a(n-1) + (n-1)!/(n-6)! * a(n-6).

a(n) = Sum_{j=0..floor(n/6)} n!/(6^j*j!*(n-6*j)!). - G. C. Greubel, Mar 07 2021