A355767
E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x*A(x)).
Original entry on oeis.org
1, 1, 0, 6, -4, 300, -828, 42224, -266992, 11916864, -132472320, 5688511488, -95465876064, 4138883728512, -95019458907072, 4276023328128000, -125481256750340352, 5958015717717504000, -212934915549001078272, 10767675634298110255104
Offset: 0
-
a(n) = sum(k=0, n, (n-2*k+1)^(k-1)*abs(stirling(n, k, 1)));
A356884
E.g.f. satisfies A(x)^A(x) = 1/(1 - x*A(x))^x.
Original entry on oeis.org
1, 0, 2, 3, 20, 150, 1254, 14280, 190000, 2863728, 49465080, 954312480, 20303200488, 473604468480, 12007399511184, 328671680500800, 9663415159357440, 303695188102656000, 10159173955921651776, 360424299614544829440, 13517056067747847719040
Offset: 0
-
a(n) = n!*sum(k=0, n\2, (n-2*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
A357422
E.g.f. satisfies A(x) * exp(A(x)) = -log(1 - x * exp(A(x))).
Original entry on oeis.org
0, 1, 1, 5, 34, 324, 3936, 58190, 1014056, 20354544, 462472800, 11733507312, 328809013776, 10086567702288, 336184985751720, 12097485061713480, 467445074411402496, 19303428522591336960, 848420150154305711616, 39543441411041750547648
Offset: 0
-
a(n) = sum(k=1, n, (n-k)^(k-1)*abs(stirling(n, k, 1)));
A356926
E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^exp(x).
Original entry on oeis.org
1, 1, 2, 3, 10, 35, 121, 1092, 5216, 39321, 558643, 2433508, 48144944, 688652549, 2176310995, 145742587616, 1334993574032, 5551320939809, 799648465754835, 1049695714507276, 90069170433616208, 6281942689646504501, -53282051261767839293, 2356158301117802408472
Offset: 0
-
nmax = 23; A[_] = 1;
Do[A[x_] = ((1 - x)^(-Exp[x]))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-exp(x)*log(1-x))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-exp(x)*log(1-x)))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*log(1-x)/lambertw(-exp(x)*log(1-x))))
A357092
E.g.f. satisfies A(x)^A(x) = (1 - x * A(x))^log(1 - x * A(x)).
Original entry on oeis.org
1, 0, 2, 6, 58, 580, 7568, 119448, 2195772, 46413792, 1106667072, 29403619080, 861570383232, 27600893313552, 959793100481616, 36006430081497120, 1449539553826089360, 62334045415459189248, 2851721291051846833152, 138299011223141244621024
Offset: 0
-
a(n) = sum(k=0, n\2, (2*k)!*(n-k+1)^(k-1)*abs(stirling(n, 2*k, 1))/k!);
A357093
E.g.f. satisfies A(x)^A(x) = 1/(1 - x * A(x))^(log(1 - x * A(x))^2).
Original entry on oeis.org
1, 0, 0, 6, 36, 210, 3150, 55104, 890232, 16735944, 386223120, 9790441056, 265867900056, 7943197796352, 260063260578576, 9156071916788544, 344740627648393920, 13880862578534022720, 595178180505073088640, 27035591386823290224000
Offset: 0
-
a(n) = sum(k=0, n\3, (3*k)!*(n-k+1)^(k-1)*abs(stirling(n, 3*k, 1))/k!);
A357094
E.g.f. satisfies A(x)^A(x) = (1 - x * A(x))^(log(1 - x * A(x)) / 2).
Original entry on oeis.org
1, 0, 1, 3, 20, 170, 1789, 22869, 342222, 5874840, 113865786, 2459446440, 58588151148, 1526055579828, 43149414029604, 1316279791377810, 43090904609439900, 1506889769163738432, 56062825134853664328, 2211097753021838716116, 92149286987928381312972
Offset: 0
-
a(n) = sum(k=0, n\2, (2*k)!*(n-k+1)^(k-1)*abs(stirling(n, 2*k, 1))/(2^k*k!));
A357095
E.g.f. satisfies A(x)^A(x) = 1/(1 - x * A(x))^(log(1 - x * A(x))^2 / 6).
Original entry on oeis.org
1, 0, 0, 1, 6, 35, 275, 2884, 35672, 494724, 7673670, 132896676, 2544253426, 53252983992, 1208888367596, 29592833903424, 777311220788320, 21808542026480120, 650880782773059840, 20590135175285212800, 688212821908314587880, 24235789570607605377680
Offset: 0
-
a(n) = sum(k=0, n\3, (3*k)!*(n-k+1)^(k-1)*abs(stirling(n, 3*k, 1))/(6^k*k!));