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a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * |Stirling1(n,3*k+2)|.
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%I #14 Oct 15 2022 08:08:52

%S 0,0,1,3,11,52,304,2114,16992,154626,1568706,17535108,213965520,

%T 2828584824,40259041188,613656673476,9971942784132,172071391424832,

%U 3141974627361216,60523400730707208,1226519845766281008,26084378634267048984,580854626450078463000

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.

%F 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(-2^(1/3) * log(1-x))/(2^(2/3)).

%F a(n) = ( (2^(1/3))_n + w * (2^(1/3)*w)_n + w^2 * (2^(1/3)*w^2)_n )/(3*2^(2/3)), where (x)_n is the Pochhammer symbol.

%o (PARI) a(n) = sum(k=0, (n-2)\3, 2^k*abs(stirling(n, 3*k+2, 1)));

%o (PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k+2)/(3*k+2)!))))

%o (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);

%o a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Pochhammer(v, n)+w*Pochhammer(v*w, n)+w^2*Pochhammer(v*w^2, n))/(3*v^2));

%Y Cf. A357831, A357832.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 14 2022