A357024
E.g.f. satisfies log(A(x)) = (exp(x * A(x)) - 1)^2.
Original entry on oeis.org
1, 0, 2, 6, 74, 750, 11402, 195006, 3994202, 93164910, 2455754762, 72098755806, 2333497474970, 82569245246670, 3170700672801482, 131342693516044926, 5837883571730770778, 277151780512413426990, 13997018265350140886282, 749304617892345721184286
Offset: 0
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m = 20; (* number of terms *)
A[_] = 0;
Do[A[x_] = Exp[(Exp[x*A[x]] - 1)^2] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)
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a(n) = sum(k=0, n\2, (2*k)!*(n+1)^(k-1)*stirling(n, 2*k, 2)/k!);
A357032
E.g.f. satisfies log(A(x)) = (exp(x * A(x)) - 1)^3 / 6.
Original entry on oeis.org
1, 0, 0, 1, 6, 25, 160, 1981, 24906, 295625, 4044900, 68136541, 1260048086, 24330807865, 508029259920, 11686882860381, 289532464998146, 7588430921962825, 210991834698749020, 6244230552027963901, 195584639712483465486, 6442981074293371848185
Offset: 0
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a(n) = sum(k=0, n\3, (3*k)!*(n+1)^(k-1)*stirling(n, 3*k, 2)/(6^k*k!));
A357010
E.g.f. satisfies log(A(x)) = (exp(x) - 1)^3 * A(x).
Original entry on oeis.org
1, 0, 0, 6, 36, 150, 1620, 24486, 293076, 3843510, 68254740, 1311687366, 25479935316, 552545882070, 13437670215060, 345157499363046, 9370414233900756, 274413997443811830, 8572526271218671380, 281754864204797848326, 9767868351458229261396
Offset: 0
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nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(Exp[x] - 1)^3*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\3, (3*k)!*(k+1)^(k-1)*stirling(n, 3*k, 2)/k!);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(exp(x)-1)^(3*k)/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-(exp(x)-1)^3))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-(exp(x)-1)^3)/(exp(x)-1)^3))
A357085
E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1)^3 * A(x).
Original entry on oeis.org
1, 0, 0, 6, 36, 150, 3780, 77406, 1059156, 21669990, 640319940, 16622025486, 450085011156, 15416323450710, 561938117029380, 20587443165165246, 835816881563118036, 37282225483118856390, 1722621978491064495300, 83817942806509377794286
Offset: 0
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a(n) = sum(k=0, n\3, (3*k)!*(n+k+1)^(k-1)*stirling(n, 3*k, 2)/k!);
Showing 1-4 of 4 results.