A356785
E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x).
Original entry on oeis.org
1, 0, 2, 3, 64, 365, 7356, 85687, 1920752, 34821369, 905128300, 22172123171, 672107454888, 20552960420005, 721088019634724, 26257726364294895, 1053711696230404576, 44336326818388565105, 2010106841636689325532, 95747319823049127621019
Offset: 0
-
nmax = 19; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x*A[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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a(n) = n!*sum(k=0, n\2, (n+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A349560
E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * x.
Original entry on oeis.org
1, 0, 2, 3, 40, 245, 2976, 35287, 524560, 8790777, 165530800, 3493679651, 80812685064, 2049413147509, 56294089065592, 1668771901644135, 53057068616526496, 1801519375618579313, 65063987978980974048, 2490449984485154892235, 100716775979173952155480
Offset: 0
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a:= n-> n!*coeff(series(RootOf(A=exp(x*exp(x*A)-x), A), x, n+1), x, n):
seq(a(n), n=0..20); # Alois P. Heinz, Nov 22 2021
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nmax = 20; A[] = 0; Do[A[x] = Exp[(E^(x*A[x]) - 1)*x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)
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my(A=1,n=22); for(i=1, n, A=exp((exp(x*A)-1)*(x+x*O(x^n)))); Vec(serlaplace(A))
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a(n) = n!*sum(k=0, n\2, (n-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Aug 27 2022
A356789
E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x)^3.
Original entry on oeis.org
1, 0, 2, 3, 112, 605, 22596, 254527, 10166416, 188035353, 8190917380, 217293592571, 10408915205976, 363500829796117, 19203682103461324, 833182131498018135, 48525371633295259936, 2511705297938365594289, 160874324235464440678164
Offset: 0
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a(n) = n!*sum(k=0, n\2, (n+2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A356797
E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^2.
Original entry on oeis.org
1, 0, 2, 3, 64, 305, 6936, 64897, 1645008, 24290289, 692240680, 14243244521, 456748635432, 12105737521033, 435619742434800, 14112089558682585, 567134312211275296, 21653262317886286817, 966207399513747354072, 42358800314758614030505
Offset: 0
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m = 20; (* number of terms *)
CoefficientList[Exp[-(1/2)*LambertW[-2*(Exp[x]-1)*x]] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 11 2022 *)
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a(n) = n!*sum(k=0, n\2, (2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*(1-exp(x)))/2)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*(1-exp(x)))/(2*x*(1-exp(x))))^(1/2)))
A356882
E.g.f. satisfies: A(x) * log(A(x)) = x * (exp(x*A(x)) - 1).
Original entry on oeis.org
1, 0, 2, 3, 16, 125, 756, 7567, 85968, 994905, 14373460, 225366251, 3800667960, 72169966453, 1469546796732, 32150706096615, 760806334538656, 19142440567996721, 512272692571487652, 14560087915617858883, 436598686303562722440, 13796641165956117509901
Offset: 0
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a(n) = n!*sum(k=0, n\2, (n-2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A356883
E.g.f. satisfies: A(x)^2 * log(A(x)) = x * (exp(x*A(x)) - 1).
Original entry on oeis.org
1, 0, 2, 3, -8, 5, 696, 2527, -40144, -178407, 8337880, 76134971, -1781542344, -24938260763, 691630553264, 14216543752335, -312910463346464, -9343318015483471, 195539694928047144, 8145971436703039363, -142317653823753257560, -8498984155838272275459
Offset: 0
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a(n) = n!*sum(k=0, n\2, (n-3*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
A371023
E.g.f. satisfies log(A(x)) = x*A(x)^2 * (exp(x*A(x)^2) - 1).
Original entry on oeis.org
1, 0, 2, 3, 112, 665, 23016, 292957, 10710960, 223877313, 9010822600, 266949248621, 12012620436312, 461111201730049, 23286625765980864, 1093225826724243045, 61822510319788946656, 3415325919719802626177, 215162865022831595415576
Offset: 0
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a(n) = n!*sum(k=0, n\2, (2*n+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
Showing 1-7 of 7 results.