A356752
E.g.f. satisfies A(x) = 1/(1 - x)^(x^2/2 * A(x)).
Original entry on oeis.org
1, 0, 0, 3, 6, 20, 360, 2394, 17220, 260280, 3076920, 35980560, 595686960, 9760411440, 159321570408, 3093987619800, 63314740616400, 1318245318411840, 30240056863978560, 736919729169603840, 18522487833889334400, 495842871278901363840, 14014346231616983128800
Offset: 0
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nmax = 22; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^2/2*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))/(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(lambertw(x^2/2*log(1-x))/(x^2/2*log(1-x))))
A356952
E.g.f. satisfies log(A(x)) = x^3/6 * (exp(x) - 1) * A(x).
Original entry on oeis.org
1, 0, 0, 0, 4, 10, 20, 35, 1736, 15204, 88320, 415965, 7632460, 121801966, 1368227224, 12184672955, 176889193040, 3490851044360, 59703361471296, 837948141904569, 13407228541467540, 283596013866706450, 6226093732482731800, 121326684752194084471
Offset: 0
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nmax = 23; A[_] = 1;
Do[A[x_] = Exp[x^3/6*(Exp[x] - 1)*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) = n!*sum(k=0, n\4, (k+1)^(k-1)*stirling(n-3*k, k, 2)/(6^k*(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^3/6*(exp(x)-1))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3/6*(1-exp(x))))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3/6*(1-exp(x)))/(x^3/6*(1-exp(x)))))
A356968
E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^3/6 * A(x)).
Original entry on oeis.org
1, 0, 0, 0, 4, 10, 40, 210, 4144, 40320, 409800, 4527600, 72552480, 1170449280, 19489513680, 338983444800, 6672681818880, 141166715289600, 3149324442700800, 73497460049395200, 1825098639493104000, 47984287767342796800, 1326460667797094860800
Offset: 0
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a(n) = n!*sum(k=0, n\4, (n-2*k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));
A356913
E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3/6).
Original entry on oeis.org
1, 0, 0, 0, 4, 10, 40, 210, 784, 5040, 40200, 369600, 5285280, 72072000, 1006889520, 14760345600, 210510263040, 3131345817600, 49229619129600, 818940523564800, 15054020163619200, 301204611031564800, 6455999452413772800, 146587705490513548800
Offset: 0
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nmax = 23; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^3/6))^(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))/(6^k*(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^3/6*log(1-x))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3/6*log(1-x)))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3/6*log(1-x)/lambertw(-x^3/6*log(1-x))))
A356925
E.g.f. satisfies A(x) = 1/(1 - x)^(exp(x) * A(x)).
Original entry on oeis.org
1, 1, 6, 51, 614, 9655, 188209, 4389532, 119363488, 3711190881, 129932611723, 5060364817200, 217054300138136, 10168837756846145, 516709033266165479, 28306732060349788908, 1663231006737554997168, 104344911495734048046929
Offset: 0
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nmax = 20; CoefficientList[Series[LambertW[E^x * Log[1-x]]/(E^x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 14 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-exp(x)*log(1-x))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(exp(x)*log(1-x)))))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(lambertw(exp(x)*log(1-x))/(exp(x)*log(1-x))))
Showing 1-5 of 5 results.