A118934 E.g.f.: exp(x + x^4/4).

1, 1, 1, 1, 7, 31, 91, 211, 1681, 12097, 57961, 209881, 1874071, 17842111, 117303187, 575683291, 5691897121, 65641390081, 544238393041, 3362783785777, 36455473647271, 485442581801311, 4828464958268491, 35900587138847971, 423276450114749617, 6318491163509870401

Offset: 0

Author

Paul D. Hanna, May 06 2006

Keywords

• nonn

Comments

Equals row sums of triangle A118933.

These are the telephone numbers T^(4)n of [Artioli et al., p. 7]. - _Eric M. Schmidt, Oct 12 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
• Vaclav Kotesovec, Graph - asymptotic (20000 terms)

Crossrefs

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).

Cf. A118933.

Programs

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

Mathematica

With[{nn=30},CoefficientList[Series[Exp[x+x^4/4],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Jan 26 2013 *)
Table[Sum[n!/(4^k*k!*(n-4*k)!), {k,0,n/4}], {n,0,30}]

PARI

a(n)=if(n<0,0,if(n==0,1,a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4)))

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

Formula

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.

a(n) ~ 1/2 * n^(3*n/4) * exp(n^(1/4)-3*n/4). - Vaclav Kotesovec, Feb 25 2014

a(n) = Sum_{k=0..floor(n/4)} n!/(4^k*k!*(n-4*k)!). - G. C. Greubel, Mar 07 2021