A356912
E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2/2).
Original entry on oeis.org
1, 0, 0, 3, 6, 20, 0, -126, -1260, 18360, 335160, 4546080, 26302320, -59501520, -5703994296, -58549768200, 371346066000, 34962417322560, 746101280831040, 8059680118183680, -93772611412099200, -5613314502242643840, -110940169654432087200
Offset: 0
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nmax = 22; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^2/2))^(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\3, (-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(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^2/2*log(1-x))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^2/2*log(1-x)))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^2/2*log(1-x)/lambertw(-x^2/2*log(1-x))))
A355287
E.g.f. satisfies A(x) = 1/(1 - x)^(x^2 * A(x)).
Original entry on oeis.org
1, 0, 0, 6, 12, 40, 1260, 8568, 62160, 1473120, 19111680, 232626240, 5403451680, 103176028800, 1822033204992, 45916616592000, 1129459815993600, 26346457488798720, 749439127417466880, 22165051763204582400, 640916967497214643200, 20787453048015928350720
Offset: 0
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nmax = 21; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^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) = n!*sum(k=0, n\3, (k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(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*log(1-x))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2*log(1-x)))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2*log(1-x))/(x^2*log(1-x))))
A356911
E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3).
Original entry on oeis.org
1, 0, 0, 0, 24, 60, 240, 1260, -12096, -120960, -1144800, -11642400, 190270080, 4670265600, 81378198720, 1348668921600, -880532674560, -406217626214400, -13255586359142400, -343166884178227200, -3137937973466572800, 72862796986940620800
Offset: 0
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nmax = 21; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^3))^(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\4, (-k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^3*log(1-x))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3*log(1-x)))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3*log(1-x)/lambertw(-x^3*log(1-x))))
Showing 1-3 of 3 results.