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Every fifth Bell number A000110.
0

%I #21 Jun 22 2022 14:44:50

%S 1,52,115975,1382958545,51724158235372,4638590332229999353,

%T 846749014511809332450147,281600203019560266563340426570,

%U 157450588391204931289324344702531067

%N Every fifth Bell number A000110.

%F a(n) = A000110(5*n).

%F a(n) = exp(-1)*Sum_{k>=0} k^(5n)/k!.

%F E.g.f.: exp(x*(d_z)^5)*(exp(exp(z)-1)) |_{z=0}, with the derivative operator d_z := d/dz. Adapted from eqs. (14) and (15) of the 1999 C. M. Bender reference given in A000110.

%t BellB[Range[0,40,5]] (* _Harvey P. Dale_, Oct 28 2019 *)

%o (PARI) for(n=0,50,print1(round(sum(i=0,1000,i^(5*n)/(i)!)/exp(1)),","))

%o (Sage) [bell_number(5*n) for n in range(0, 9)] # _Zerinvary Lajos_, May 15 2009

%o (Python)

%o from itertools import accumulate, islice

%o def A070908_gen(): # generator of terms

%o yield 1

%o blist, b = (1,), 1

%o while True:

%o for _ in range(5):

%o blist = list(accumulate(blist, initial=(b:=blist[-1])))

%o yield b

%o A070908_list = list(islice(A070908_gen(),20)) # _Chai Wah Wu_, Jun 22 2022

%K easy,nonn

%O 0,2

%A _Benoit Cloitre_, May 19 2002