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A357831
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a(n) = Sum_{k=0..floor(n/3)} 2^k * |Stirling1(n,3*k)|.
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4
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1, 0, 0, 2, 12, 70, 454, 3332, 27552, 254400, 2598852, 29125932, 355455468, 4693396656, 66671326176, 1013916648840, 16436063079552, 282920894841096, 5153797995148296, 99052313167391760, 2003040751641857856, 42513854724369719136, 944959706480298199824
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OFFSET
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0,4
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LINKS
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FORMULA
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Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(-2^(1/3) * log(1-x)).
a(n) = ( (2^(1/3))_n + (2^(1/3)*w)_n + (2^(1/3)*w^2)_n )/3, where (x)_n is the Pochhammer symbol.
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PROG
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(PARI) a(n) = sum(k=0, n\3, 2^k*abs(stirling(n, 3*k, 1)));
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k)/(3*k)!)))
(PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round(Pochhammer(v, n)+Pochhammer(v*w, n)+Pochhammer(v*w^2, n))/3;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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