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A357830 a(n) = Sum_{k=0..floor((n-2)/3)} |Stirling1(n,3*k+2)|.

%I A357830 #19 Feb 16 2025 08:34:04
%S A357830 0,0,1,3,11,51,289,1939,15029,132069,1296771,14063721,166897059,
%T A357830 2150579067,29895590361,445871456667,7100686041813,120249378265653,
%U A357830 2157637558311963,40887284144179473,815949872494416387,17103401793743095467,375692072337527815233
%N A357830 a(n) = Sum_{k=0..floor((n-2)/3)} |Stirling1(n,3*k+2)|.
%H A357830 Robert Israel, <a href="/A357830/b357830.txt">Table of n, a(n) for n = 0..448</a>
%H A357830 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357830 Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(-log(1-x)).
%F A357830 a(n) = ( (1)_n + w * (w)_n + w^2 * (w^2)_n )/3, where (x)_n is the Pochhammer symbol.
%p A357830 f:= proc(n) local k;  add(abs(Stirling1(n,3*k+2)), k=0..(n-2)/3) end proc:
%p A357830 map(f, [$0..30]); # _Robert Israel_, Feb 12 2024
%t A357830 Table[Sum[Abs[StirlingS1[n,3k+2]],{k,0,Floor[(n-2)/3]}],{n,0,30}] (* _Harvey P. Dale_, Jan 12 2024 *)
%o A357830 (PARI) a(n) = sum(k=0, (n-2)\3, abs(stirling(n, 3*k+2, 1)));
%o A357830 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, (-log(1-x))^(3*k+2)/(3*k+2)!))))
%o A357830 (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o A357830 a(n) = my(w=(-1+sqrt(3)*I)/2); round(Pochhammer(1, n)+w*Pochhammer(w, n)+w^2*Pochhammer(w^2, n))/3;
%Y A357830 Cf. A357828, A357829.
%K A357830 nonn
%O A357830 0,4
%A A357830 _Seiichi Manyama_, Oct 14 2022