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Every fourth Bell number A000110.
0

%I #24 Jun 22 2022 14:44:46

%S 1,15,4140,4213597,10480142147,51724158235372,445958869294805289,

%T 6160539404599934652455,128064670049908713818925644,

%U 3819714729894818339975525681317,157450588391204931289324344702531067

%N Every fourth Bell number A000110.

%F a(n) = A000110(4*n).

%F a(n) = exp(-1)*Sum_{k>=0} k^(4n)/k!.

%F E.g.f.: exp(x*(d_z)^4)*(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 Table[BellB[4*n], {n, 0, 10}] (* _Amiram Eldar_, Jun 11 2022 *)

%o (PARI) for(n=0,50,print1(round(sum(i=0,1000,i^(4*n)/(i)!)/exp(1)),","))

%o (Sage) [bell_number(4*n) for n in range(0, 11)] # _Zerinvary Lajos_, May 15 2009

%o (Python)

%o from itertools import accumulate, islice

%o def A070907_gen(): # generator of terms

%o yield 1

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

%o while True:

%o for _ in range(4):

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

%o yield b

%o A070907_list = list(islice(A070907_gen(),20)) # _Chai Wah Wu_, Jun 22 2022

%Y Cf. A000110.

%K easy,nonn

%O 0,2

%A _Benoit Cloitre_, May 19 2002