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A344559 a(n) = (1/6) * 2^(-n) * n! * [x^n] Exp(2*x, 1)*(Exp(2*x, 3) - 1), where Exp(x, m) = Sum_{k>=0} (x^k / k!)^m.

%I A344559 #9 Jan 04 2024 14:04:14
%S A344559 0,0,0,1,4,10,35,140,476,1624,6070,22495,81455,301301,1131494,4230681,
%T A344559 15852396,59881956,226877648,860447129,3273728234,12493453344,
%U A344559 47760610689,182905145214,701883651799,2697952583635,10385325566785,40033903418860,154534663044346
%N A344559 a(n) = (1/6) * 2^(-n) * n! * [x^n] Exp(2*x, 1)*(Exp(2*x, 3) - 1), where Exp(x, m) = Sum_{k>=0} (x^k / k!)^m.
%F A344559 a(n) = A344854(n) / 2^n.
%F A344559 a(n) = (1/6)*(hypergeom([-n/3, (1 - n)/3, (2 - n)/3], [1, 1], -27) - 1).
%p A344559 Exp := (x, m) -> sum((x^k / k!)^m, k=0..infinity):
%p A344559 gf := Exp(2*x, 1)*(Exp(2*x, 3) - 1): ser := series(gf, x, 34):
%p A344559 seq((1/6)*2^(-n)*n!*simplify(coeff(ser, x, n)), n = 0..28);
%t A344559 a[n_] := (1/6) (HypergeometricPFQ[{-n/3, (1 - n)/3, (2 - n)/3}, {1, 1}, -27] - 1);
%t A344559 Table[a[n], {n, 0, 28}]
%o A344559 (Python)
%o A344559 from sympy import hyperexpand, Rational
%o A344559 from sympy.functions import hyper
%o A344559 def A344559(n): return (hyperexpand(hyper((Rational(-n,3),Rational(1-n,3),Rational(2-n,3)),(1,1),-27))-1)//6 # _Chai Wah Wu_, Jan 04 2024
%Y A344559 Cf. A344854.
%K A344559 nonn
%O A344559 0,5
%A A344559 _Peter Luschny_, Jun 01 2021