A356798
E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^3.
Original entry on oeis.org
1, 0, 2, 3, 88, 425, 13476, 130417, 4543120, 71005041, 2723297860, 60685651961, 2564091428856, 75166650583609, 3496499475113932, 127585829832674865, 6521845096842043936, 284745004488498858209, 15950013722559213419412, 809403234909367349670409
Offset: 0
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nmax = 19; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x] - 1)*A[x]^3] + 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) = n!*sum(k=0, n\2, (3*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, (3*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(3*x*(1-exp(x)))/3)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*(1-exp(x)))/(3*x*(1-exp(x))))^(1/3)))
A356902
E.g.f. satisfies A(x) * log(A(x)) = x * (exp(x) - 1).
Original entry on oeis.org
1, 0, 2, 3, -8, -55, 276, 4417, -13488, -639567, -248300, 141842921, 797525400, -43103642855, -584650622724, 16366430341185, 436555007091616, -6909610676492959, -368240758971238620, 2371795171252419385, 354876368637537736680, 1050192150132691993161
Offset: 0
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nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(Exp[x] - 1)*x/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, (-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*(exp(x)-1)))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*(1-exp(x))/lambertw(-x*(1-exp(x)))))
A356949
E.g.f. satisfies log(A(x)) = x^2 * (exp(x) - 1) * A(x).
Original entry on oeis.org
1, 0, 0, 6, 12, 20, 1110, 7602, 35336, 1103832, 14984010, 134552990, 3457329612, 70828191876, 1017237973934, 25648737955050, 676111332667920, 13760810592066992, 373071111301807506, 11594147432172228918, 307097278689726728660, 9330499711181779575900
Offset: 0
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nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(-1 + Exp[x])*A[x]*x^2] + 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\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^2*(exp(x)-1))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2*(1-exp(x))))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2*(1-exp(x)))/(x^2*(1-exp(x)))))
A356904
E.g.f. satisfies A(x)^2 * log(A(x)) = x * (exp(x) - 1).
Original entry on oeis.org
1, 0, 2, 3, -32, -175, 2376, 29617, -371440, -9251919, 91421560, 4529155961, -26677647864, -3160004989271, 1541460644192, 2946529440977865, 19556193589426336, -3498019439220155551, -56274505323609293208, 5077223330715030358009
Offset: 0
-
nmax = 19; A[_] = 1;
Do[A[x_] = Exp[((Exp[x]-1)*x)/A[x]^2] + 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) = 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*(exp(x)-1))/2)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*(1-exp(x))/lambertw(-2*x*(1-exp(x))))^(1/2)))
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
-
a(n) = n!*sum(k=0, n\2, (2*n+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
Showing 1-5 of 5 results.