A356753
E.g.f. satisfies A(x) = 1/(1 - x)^(x^3/6 * A(x)).
Original entry on oeis.org
1, 0, 0, 0, 4, 10, 40, 210, 3024, 25200, 225000, 2217600, 29974560, 400720320, 5558957040, 81340459200, 1344965825280, 23566775232000, 432681781459200, 8309927446329600, 170258024427580800, 3679448236206220800, 83235946152090547200, 1962840630226968307200
Offset: 0
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nmax = 23; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^3/6*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(lambertw(x^3/6*log(1-x))/(x^3/6*log(1-x))))
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))))
A356951
E.g.f. satisfies log(A(x)) = x^2/2 * (exp(x) - 1) * A(x).
Original entry on oeis.org
1, 0, 0, 3, 6, 10, 285, 1911, 8848, 155016, 1931625, 17006275, 276807036, 4801114968, 65672925409, 1172625764415, 24657199159440, 460156401399376, 9560083801337793, 230955040794126915, 5393971086379904260, 131545127670380245920, 3587507216606547324321
Offset: 0
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nmax = 22; A[_] = 1;
Do[A[x_] = Exp[x^2/2*(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\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(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*(exp(x)-1))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2/2*(1-exp(x))))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2/2*(1-exp(x)))/(x^2/2*(1-exp(x)))))
A356967
E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^2/2 * A(x)).
Original entry on oeis.org
1, 0, 0, 3, 6, 20, 540, 4284, 35700, 698760, 10591560, 155024100, 3231336240, 68704908480, 1451024062488, 35843008924800, 950925111973200, 25881184257364800, 765471782499860160, 24210772652940214320, 797021694060934204800, 27752674398657594805440
Offset: 0
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a(n) = n!*sum(k=0, n\3, (n-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));
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.