A277489
Expansion of e.g.f. -LambertW(-log(1+x)).
Original entry on oeis.org
0, 1, 1, 5, 26, 224, 2244, 28496, 417976, 7122384, 136770960, 2937770472, 69626588976, 1806936836184, 50936933449752, 1550292926398680, 50661309325357824, 1769286989373994752, 65762170385201959680, 2591979585702305271552, 107982615297265761991680
Offset: 0
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CoefficientList[Series[-LambertW[-Log[1+x]], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[StirlingS1[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]
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my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-log(1+x))))) \\ G. C. Greubel, Jun 21 2017
A357337
E.g.f. satisfies A(x) = log(1 + x) * exp(2 * A(x)).
Original entry on oeis.org
0, 1, 3, 26, 334, 5964, 135228, 3729872, 121172560, 4532603904, 191869653120, 9067948437888, 473297792213376, 27039987154142208, 1678363256057198592, 112467293224249912320, 8092242817059530284032, 622253112192770288799744
Offset: 0
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nmax = 20; A[_] = 0;
Do[A[x_] = Log[1 + x]*Exp[2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
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my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*log(1+x))/2)))
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a(n) = sum(k=1, n, (2*k)^(k-1)*stirling(n, k, 1));
A357392
E.g.f. satisfies A(x) = -log(1 - x * exp(2 * A(x))).
Original entry on oeis.org
0, 1, 5, 56, 990, 24024, 742560, 27907200, 1235591280, 62990928000, 3634245014400, 234102016512000, 16654322805120000, 1296884927852236800, 109720581991308288000, 10021650950985427353600, 982869376029609100032000, 103017324974226408345600000
Offset: 0
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a(n) = sum(k=1, n, (2*n)^(k-1)*abs(stirling(n, k, 1)));
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a(n) = sum(k=1, n, (3*n)^(k-1)*stirling(n, k, 1));
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a(n) = if(n==0, 0, (3*n-1)!/(2*n)!);
Showing 1-3 of 3 results.