A349504
E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).
Original entry on oeis.org
1, 1, 4, 36, 484, 8840, 203868, 5691308, 186612592, 7031373264, 299397454080, 14218443479328, 745142534904480, 42717896158340832, 2659373970144454080, 178666030775042040000, 12884568940594969258752, 992750028716940749121792
Offset: 0
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nmax = 20; CoefficientList[Series[Sqrt[-LambertW[-2*Log[1 + x]]/(2*Log[1 + x])], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 20 2021 *)
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a(n) = sum(k=0,n, (2*k+1)^(k-1)*stirling(n, k, 1));
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N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*log(1+x)^k/k!)))
A264408
E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^3).
Original entry on oeis.org
1, 1, 8, 123, 2880, 91380, 3670422, 178637046, 10220188584, 672297233904, 49998246211440, 4148400045354048, 379922492043352728, 38069325641907527760, 4143052833221549219112, 486645149764644576491160, 61364015736691449905270208, 8267876533375590035644494336, 1185420892076469924434688605760
Offset: 0
E.g.f.: A(x) = 1 + x + 8*x^2/2! + 123*x^3/3! + 2880*x^4/4! + 91380*x^5/5! + 3670422*x^6/6! + 178637046*x^7/7! + 10220188584*x^8/8! +...
where A(x) = 1/(1-x)^(A(x)^3).
From a LambertW identity,
A(x) = 1 - log(1-x) + 7*log(1-x)^2/2! - 10^2*log(1-x)^3/3! + 13^3*log(1-x)^4/4! - 16^4*log(1-x)^5/5! + 19^5*log(1-x)^6/6! +...
Also,
A(x) = 1 + x*A(x)^3 + x^2*A(x)^3*(A(x)^3+1)/2! + x^3*A(x)^3*(A(x)^3+1)*(A(x)^3+2)/3! + x^4*A(x)^3*(A(x)^3+1)*(A(x)^3+2)*(A(x)^3+3)/4! +...
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Table[Sum[Abs[StirlingS1[n, k]] * (3*k+1)^(k-1), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 18 2015 *)
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{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A^3 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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a(n) = sum(k=0, n, abs(stirling(n, k, 1))*(3*k+1)^(k-1));
for(n=0, 20, print1(a(n), ", "))
A357333
E.g.f. satisfies A(x) = -log(1 - x) * exp(2 * A(x)).
Original entry on oeis.org
0, 1, 5, 50, 778, 16604, 451668, 14947568, 582982160, 26187136128, 1331445995520, 75589772147328, 4739901861071232, 325353447339098112, 24264683011603485696, 1953776475810372817920, 168924939633683095452672, 15609228287753846217412608
Offset: 0
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nmax = 17; 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)*abs(stirling(n, k, 1)));
Showing 1-3 of 3 results.