A357321
Expansion of e.g.f. -LambertW(log(1 - 2*x)/2).
Original entry on oeis.org
0, 1, 4, 29, 308, 4349, 77094, 1650893, 41532280, 1201865049, 39351776970, 1438731784137, 58107225611412, 2569486856423733, 123475320944016846, 6407225728624769925, 357061085760608504304, 21268522319028809507889, 1348496822257863921774738
Offset: 0
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With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[Log[1 - 2*x]/2], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
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my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(log(1-2*x)/2))))
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a(n) = sum(k=1, n, 2^(n-k)*k^(k-1)*abs(stirling(n, k, 1)));
A357322
Expansion of e.g.f. -LambertW(log(1 - 3*x)/3).
Original entry on oeis.org
0, 1, 5, 45, 586, 10024, 213084, 5428072, 161475320, 5501761488, 211466328400, 9057714349672, 428022643010544, 22127292215218072, 1242503403120434168, 75319473068729478360, 4902798528238919060224, 341102498012848479889408
Offset: 0
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With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[Log[1 - 3*x]/3], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
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my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(log(1-3*x)/3))))
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a(n) = sum(k=1, n, 3^(n-k)*k^(k-1)*abs(stirling(n, k, 1)));
A355874
Expansion of e.g.f. -LambertW(x^2 * log(1-x))/2.
Original entry on oeis.org
0, 0, 0, 3, 6, 20, 450, 3024, 21840, 449280, 5690160, 68579280, 1491462720, 27798076800, 485405784864, 11821894207200, 285057334598400, 6578025489584640, 183420564173141760, 5342163886869062400, 152988752430721267200, 4897735504358795965440
Offset: 0
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With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^2 * Log[1 - x]]/2, {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
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my(N=20, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2*log(1-x)))/2))
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a(n) = n!*sum(k=1, n\3, k^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/2;
A355993
Expansion of e.g.f. -LambertW(x^3 * log(1-x))/6.
Original entry on oeis.org
0, 0, 0, 0, 4, 10, 40, 210, 8064, 70560, 640800, 6375600, 189383040, 3165402240, 48879754560, 762766804800, 21652937349120, 525738717504000, 11796584629939200, 259139188966694400, 7842638783736115200, 240231375437935795200, 7066934411387842252800
Offset: 0
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With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^3 * Log[1 - x]]/6, {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
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my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3*log(1-x)))/6))
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a(n) = n!*sum(k=1, n\4, k^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!)/6;
A355994
Expansion of e.g.f. -LambertW(x^2/2 * log(1-x)).
Original entry on oeis.org
0, 0, 0, 3, 6, 20, 270, 1764, 12600, 169560, 1937880, 22300740, 349806600, 5556245760, 89073856872, 1678920566400, 33550354656000, 687175528253760, 15462823882213440, 370285712520237360, 9180722384533375200, 242398467521271149760
Offset: 0
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With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^2/2 * Log[1 - x]], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
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my(N=20, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2/2*log(1-x)))))
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a(n) = n!*sum(k=1, n\3, k^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));
A355995
Expansion of e.g.f. -LambertW(x^3/6 * log(1-x)).
Original entry on oeis.org
0, 0, 0, 0, 4, 10, 40, 210, 2464, 20160, 178800, 1755600, 22323840, 289729440, 3950069760, 57127870800, 921032555520, 15786602832000, 284810759251200, 5394363163862400, 108742028591923200, 2312415679065811200, 51543520889668684800, 1199641884471310156800
Offset: 0
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With[{m = 25}, Range[0, m]! * CoefficientList[Series[-ProductLog[x^3/6 * Log[1 - x]], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
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my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3/6*log(1-x)))))
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a(n) = n!*sum(k=1, n\4, k^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));