A356363
a(n) = Sum_{k=0..floor(n/3)} n^k * Stirling2(n,3*k).
Original entry on oeis.org
1, 0, 0, 3, 24, 125, 576, 3136, 24752, 242280, 2421000, 23568743, 230156136, 2370756505, 26664718080, 326641069815, 4243004068192, 57065900282730, 787656999701016, 11193821784313606, 165023822310642520, 2535785869709189307, 40583218821499596176
Offset: 0
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a(n) = sum(k=0, n\3, n^k*stirling(n, 3*k, 2));
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a(n) = n!*polcoef(sum(k=0, n\3, n^k*(exp(x+x*O(x^n))-1)^(3*k)/(3*k)!), n);
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Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(v=n^(1/3), w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, v)+Bell_poly(n, v*w)+Bell_poly(n, v*w^2))/3;
A357783
a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * Stirling2(n,3*k+1).
Original entry on oeis.org
0, 1, 1, 1, 3, 21, 131, 705, 3515, 17389, 91739, 547889, 3746227, 28241373, 224124083, 1821051233, 15023818091, 126366334125, 1094358852075, 9858890038513, 92983173940419, 918408372280477, 9454438841355395, 100728532687727329, 1103649166937235259
Offset: 0
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a(n) = sum(k=0, (n-1)\3, 2^k*stirling(n, 3*k+1, 2));
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N\3, 2^k*(exp(x)-1)^(3*k+1)/(3*k+1)!))))
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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^2*Bell_poly(n, v*w)+w*Bell_poly(n, v*w^2))/(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.