A349583
E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.
Original entry on oeis.org
1, 1, 0, 2, -9, 72, -710, 8563, -121814, 1997502, -37097739, 769687954, -17644355410, 442894514285, -12081668234012, 355889274553166, -11258683640579857, 380701046875217492, -13702507978018209458, 523049811008797507683, -21105565578064063658754
Offset: 0
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b:= proc(n, m) option remember; `if`(n=0,
(1-m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..24); # Alois P. Heinz, Jul 29 2022
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a[n_] := Sum[If[k == 1, 1, (-k + 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 23 2021 *)
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a(n) = sum(k=0, n, (-k+1)^(k-1)*stirling(n, k, 2));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(exp(x)-1))))
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))
A349561
E.g.f. satisfies: A(x)^A(x) = 1/(1 - x).
Original entry on oeis.org
1, 1, 0, 3, -8, 100, -834, 11438, -159928, 2762352, -52322160, 1124320032, -26509832040, 686751503568, -19306448087640, 586539826169880, -19131996548499264, 667157522614934016, -24762890955027112128, 974824890777753840576, -40566428716555791936000
Offset: 0
A(x) - 1 = x + 3*x^3/6 - 8*x^4/24 + ... = x + x^3/2 - x^4/3 + ... .
A(x)^A(x) = (1 + (A(x) - 1))^(1 + (A(x) - 1)) = Sum_{k>=0} A005727(k) * (A(x) - 1)^k / k! = 1 + 1 * (x + x^3/2 - x^4/3 + ... )/1! + 2 * (x + x^3/2 - x^4/3 + ... )^2/2! + 3 * (x + x^3/2 - x^4/3 + ... )^3/3! + ... = 1 + x + x^2 + x^3 + ... = 1/(1 - x).
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Join[{1}, Table[(n-1)! - (-1)^n*Sum[(k-1)^(k-1)*StirlingS1[n, k], {k, 2, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 22 2021 *)
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a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (k-1)^(k-1)*log(1-x)^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1-x)/lambertw(-log(1-x))))
A349585
E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).
Original entry on oeis.org
1, 1, -2, 8, -59, 642, -9112, 158839, -3279880, 78250188, -2117569181, 64082989720, -2144319848772, 78609355884893, -3133061858717806, 134884905211588892, -6238095343894356675, 308427209934965151158, -16234730389499986865092, 906409067599064528054343
Offset: 0
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b:= proc(n, m) option remember; `if`(n=0,
(m-1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> (-1)^(n-1)*b(n, 0):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 03 2022
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a[n_] := (-1)^(n - 1) * Sum[If[k == 1, 1, (k - 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 23 2021 *)
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a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 2));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(1-exp(-x)))))
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my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))
A349587
E.g.f. satisfies: A(x)^A(x) = 1 + x*A(x).
Original entry on oeis.org
1, 1, 0, -3, 4, 60, -294, -2800, 34504, 197568, -6087360, -9146808, 1488986808, -5886157992, -469973309064, 5492298353880, 177826238321856, -4277426240130048, -72353540601814464, 3537861051231290880, 22847222673714931200, -3226666120379253611136
Offset: 0
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a[n_] := Sum[(n - k + 1)^(k - 1)*StirlingS1[n, k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Nov 23 2021 *)
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a(n) = sum(k=0, n, (n-k+1)^(k-1)*stirling(n, k, 1));
A349650
E.g.f. satisfies: A(x)^(A(x)^2) = 1 + x.
Original entry on oeis.org
1, 1, -4, 36, -532, 11040, -295188, 9655772, -373422320, 16666348464, -843095987520, 47669276120928, -2979044176833888, 203906085094788960, -15170476121142482112, 1218972837861962011200, -105202043767190506428672, 9705514148732971389369600
Offset: 0
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nmax = 20; A[_] = 1;
Do[A[x_] = (1 + x)^(1/A[x]^2) + 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) = (-1)^(n-1)*sum(k=0, n, (2*k-1)^(k-1)*abs(stirling(n, k, 1)));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (2*k-1)^(k-1)*(-log(1+x))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((2*log(1+x)/lambertw(2*log(1+x)))^(1/2)))
A349651
E.g.f. satisfies: A(x)^(A(x)^3) = 1 + x.
Original entry on oeis.org
1, 1, -6, 81, -1776, 54240, -2125122, 101631558, -5739235128, 373745355984, -27572590788480, 2272763834553168, -207013811669644680, 20647997125333476912, -2238256520486195804280, 262010379635788799196360, -32939968662220720559744448
Offset: 0
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nmax = 20; A[_] = 1;
Do[A[x_] = (1 + x)^(1/A[x]^3) + 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) = (-1)^(n-1)*sum(k=0, n, (3*k-1)^(k-1)*abs(stirling(n, k, 1)));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (3*k-1)^(k-1)*(-log(1+x))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((3*log(1+x)/lambertw(3*log(1+x)))^(1/3)))
A349600
E.g.f. satisfies: A(x)^A(x) = 1 + x*A(x)^2.
Original entry on oeis.org
1, 1, 2, 3, -20, -320, -2274, 5474, 487432, 8358480, 37944240, -2286868848, -81319780200, -1139790073968, 18382692073032, 1570867988794680, 42704382709868736, 55662087673489920, -49662902468183117760, -2360239974764654675904, -38098700311039336972800
Offset: 0
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a(n) = sum(k=0, n, (2*n-k+1)^(k-1)*stirling(n, k, 1));
Showing 1-7 of 7 results.