A356909
E.g.f. satisfies A(x)^A(x) = 1/(1 - 2*x)^(1/2).
Original entry on oeis.org
1, 1, 1, 6, 25, 260, 2031, 29414, 343729, 6331464, 99347775, 2213854962, 43775364777, 1143045179484, 27318538754703, 817488701386590, 22930078960458081, 773538392955105552, 24914603847636688383, 935548306987015373418, 34024565857537814925465
Offset: 0
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nmax = 20; A[_] = 1;
Do[A[x_] = (1 - 2*x)^(-(1/2)/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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a(n) = sum(k=0, n, 2^(n-k)*(-k+1)^(k-1)*abs(stirling(n, k, 1)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-log(1-2*x)/2)^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-log(1-2*x)/2))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-log(1-2*x)/(2*lambertw(-log(1-2*x)/2))))
A357243
E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(1 - x).
Original entry on oeis.org
1, 1, -2, 6, -52, 540, -7608, 129304, -2612608, 60867360, -1608663840, 47527158624, -1552431588288, 55547889458880, -2160724031160576, 90782738645280000, -4097139872604807168, 197675862365363088384, -10153243488783257091072
Offset: 0
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nmax = 20; A[_] = 1;
Do[A[x_] = ((1 - x)^(-1 + 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=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-(1-x)*log(1-x))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-(1-x)*log(1-x)))))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-(1-x)*log(1-x)/lambertw(-(1-x)*log(1-x))))
A357244
E.g.f. satisfies A(x) * log(A(x)) = 2 * (exp(x) - 1).
Original entry on oeis.org
1, 2, -2, 22, -266, 4614, -102442, 2777030, -88914730, 3283693254, -137408080298, 6425417730758, -332055079469610, 18792899306652358, -1156017201432075946, 76796076655220486854, -5479395288838822143786, 417905042599836811225798, -33928512587303405767179178
Offset: 0
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nmax = 18; A[_] = 1;
Do[A[x_] = Exp[(2*(Exp[x] - 1))/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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a(n) = sum(k=0, n, 2^k*(-k+1)^(k-1)*stirling(n, k, 2));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(2*(exp(x)-1))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*(exp(x)-1)))))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(2*(exp(x)-1)/lambertw(2*(exp(x)-1))))
Showing 1-3 of 3 results.