A357831
a(n) = Sum_{k=0..floor(n/3)} 2^k * |Stirling1(n,3*k)|.
Original entry on oeis.org
1, 0, 0, 2, 12, 70, 454, 3332, 27552, 254400, 2598852, 29125932, 355455468, 4693396656, 66671326176, 1013916648840, 16436063079552, 282920894841096, 5153797995148296, 99052313167391760, 2003040751641857856, 42513854724369719136, 944959706480298199824
Offset: 0
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a(n) = sum(k=0, n\3, 2^k*abs(stirling(n, 3*k, 1)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k)/(3*k)!)))
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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;
A357832
a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * |Stirling1(n,3*k+1)|.
Original entry on oeis.org
0, 1, 1, 2, 8, 44, 290, 2194, 18690, 177072, 1848048, 21079332, 260998584, 3487438476, 50030096844, 767092681992, 12520306878720, 216760973139072, 3967857438205320, 76575231882844056, 1553981718941428824, 33082675130470434336, 737250032464248840192
Offset: 0
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a[n_] := With[{v = 2^(1/3), w = (-1 + Sqrt[3]*I)/2}, Round[(Pochhammer[v, n] + w^2*Pochhammer[v*w, n] + w*Pochhammer[v*w^2, n])/(3*v)]];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Oct 16 2022, after 3rd PARI code *)
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a(n) = sum(k=0, (n-1)\3, 2^k*abs(stirling(n, 3*k+1, 1)));
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k+1)/(3*k+1)!))))
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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)+w^2*Pochhammer(v*w, n)+w*Pochhammer(v*w^2, n))/(3*v));
A357784
a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * Stirling2(n,3*k+2).
Original entry on oeis.org
0, 0, 1, 3, 7, 17, 61, 343, 2231, 14301, 88561, 542011, 3397483, 22638993, 164336085, 1299899087, 10991061663, 97070035205, 881323166809, 8173386231395, 77489746906355, 754631383660729, 7590899551399869, 79174328607339767, 856889470005396071
Offset: 0
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a(n) = sum(k=0, (n-2)\3, 2^k*stirling(n, 3*k+2, 2));
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my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, 2^k*(exp(x)-1)^(3*k+2)/(3*k+2)!))))
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Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Bell_poly(n, v)+w*Bell_poly(n, v*w)+w^2*Bell_poly(n, v*w^2))/(3*v^2));
Showing 1-3 of 3 results.