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A293588
E.g.f.: exp(x + x^6/6).
3
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
0,7
COMMENTS
These are the telephone numbers T^(6)_n of [Artioli et al., p. 7].
LINKS
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
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
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 *)
PROG
(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
(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
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).
Sequence in context: A306452 A238250 A361658 * A365969 A203959 A362319
KEYWORD
nonn
AUTHOR
Eric M. Schmidt, Oct 12 2017
STATUS
approved