A356905
E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.
Original entry on oeis.org
1, 0, 2, 3, -4, -30, 294, 3780, -7904, -444528, 78840, 99657360, 539299848, -27852945120, -361237078944, 10124338180320, 258341121976320, -4020500134465920, -205187357182405824, 1330097523844832640, 186823640933648588160, 500469438126120583680
Offset: 0
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nmax = 21; A[_] = 1;
Do[A[x_] = (1/(1 - x)^x)^(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\2, (-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(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*log(1-x))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x*log(1-x)))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*log(1-x)/lambertw(-x*log(1-x))))
A356796
E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)^3).
Original entry on oeis.org
1, 0, 2, 3, 92, 450, 14454, 141540, 4980128, 78711696, 3048567480, 68677353360, 2930551701384, 86832573553440, 4079649847428960, 150444517302424800, 7768028697749806080, 342721736137376184960, 19392702029822685015360, 994397473912386435004800
Offset: 0
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a(n) = n!*sum(k=0, n\2, (3*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(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*log(1-x))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*x*log(1-x))/3)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(3*x*log(1-x))/(3*x*log(1-x)))^(1/3)))
A356906
E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x)^x.
Original entry on oeis.org
1, 0, 2, 3, -28, -150, 2154, 26040, -322512, -7872984, 77570280, 3752301960, -22068935736, -2542757920560, 1422846762960, 2302464947491800, 14860063644794880, -2653728770258072640, -41790782141846648640, 3739260018343338345600
Offset: 0
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a(n) = n!*sum(k=0, n\2, (-2*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(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*log(1-x))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-2*x*log(1-x))/2)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*log(1-x)/lambertw(-2*x*log(1-x)))^(1/2)))
A371146
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^2)^(x*A(x)^2).
Original entry on oeis.org
1, 0, 2, 3, 116, 690, 24714, 315840, 11919088, 250812072, 10389272040, 310700914920, 14351129171400, 557402214180240, 28831564284582864, 1372162923004025880, 79345973798740154880, 4450055092134985771200, 286324089075857021558976
Offset: 0
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a(n) = n!*sum(k=0, n\2, (2*n+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
Showing 1-4 of 4 results.