A356785 -id:A356785 - cp's OEIS frontend

A355843 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x).

1, 0, 2, 3, 40, 185, 2556, 22057, 349616, 4519377, 83642860, 1439639201, 31015493928, 663158322697, 16468280168900, 418772642545545, 11847925722273376, 348085509493265825, 11091199095506163420, 368912674236287743633, 13099432280183074041560

Offset: 0

Seiichi Manyama, Jul 18 2022

nonn

Cf. A052880, A355781, A356785.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x*(1-exp(x))))))

a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Aug 28 2022

E.g.f.: exp( -LambertW(x * (1 - exp(x))) ).
E.g.f.: LambertW(x * (1 - exp(x))) / (x * (1 - exp(x))).
a(n) ~ sqrt(1 + exp(1+r)*r^2) * n^(n-1) / (exp(n-1) * r^n), where r = 0.528399250336668412340528181936966763473482889289226687323... is the root of the equation exp(1+r) - exp(1) = 1/r. - Vaclav Kotesovec, Jul 21 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. - Seiichi Manyama, Aug 28 2022

A349560 E.g.f. satisfies log(A(x)) = (exp(xA(x)) - 1) x.

Original entry on oeis.org

1, 0, 2, 3, 40, 245, 2976, 35287, 524560, 8790777, 165530800, 3493679651, 80812685064, 2049413147509, 56294089065592, 1668771901644135, 53057068616526496, 1801519375618579313, 65063987978980974048, 2490449984485154892235, 100716775979173952155480

Offset: 0

Seiichi Manyama, Nov 22 2021

nonn

Cf. A052506, A349557.

Cf. A356785, A356788, A356789.

a:= n-> n!*coeff(series(RootOf(A=exp(x*exp(x*A)-x), A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 22 2021

nmax = 20; A[] = 0; Do[A[x] = Exp[(E^(x*A[x]) - 1)*x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)

my(A=1,n=22); for(i=1, n, A=exp((exp(x*A)-1)*(x+x*O(x^n)))); Vec(serlaplace(A))

a(n) = n!*sum(k=0, n\2, (n-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!); \\ Seiichi Manyama, Aug 27 2022

a(n) ~ sqrt(s*(1 - r^2*s/(1 + r*s))) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.4599551063707173872728335298048828687860291021728... is the root of the equation r - LambertW(1/r) - 2*log(r) = 1/LambertW(1/r) and s = LambertW(1/r)/r = 1.938208283387405345404104769972407921289092368509... - Vaclav Kotesovec, Nov 22 2021

a(n) = n! * Sum_{k=0..floor(n/2)} (n-k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!. - Seiichi Manyama, Aug 27 2022

A356788 E.g.f. satisfies log(A(x)) = x * (exp(xA(x)) - 1) A(x)^2.

Original entry on oeis.org

1, 0, 2, 3, 88, 485, 13896, 158767, 4919664, 90698841, 3130084360, 81025744811, 3144372342552, 104942286748741, 4582896912897408, 186591555463556895, 9135453970592830816, 437146665470130792497, 23852990622867670807704, 1307029600226135900982835

Offset: 0

Seiichi Manyama, Aug 27 2022

nonn

Cf. A349560, A356785, A356789.

Cf. A355762.

a(n) = n!*sum(k=0, n\2, (n+k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} (n+k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.

A356789 E.g.f. satisfies log(A(x)) = x * (exp(xA(x)) - 1) A(x)^3.

Original entry on oeis.org

1, 0, 2, 3, 112, 605, 22596, 254527, 10166416, 188035353, 8190917380, 217293592571, 10408915205976, 363500829796117, 19203682103461324, 833182131498018135, 48525371633295259936, 2511705297938365594289, 160874324235464440678164

Offset: 0

Seiichi Manyama, Aug 27 2022

nonn

Cf. A349560, A356785, A356788.

a(n) = n!*sum(k=0, n\2, (n+2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} (n+2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.

A356962 E.g.f. satisfies log(A(x)) = x^2/2 * (exp(xA(x)) - 1) A(x).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 465, 3801, 20608, 461196, 7609185, 85446955, 1661943756, 38070386718, 692342989429, 15023805426735, 404978989779120, 10131679290423736, 264474729910772433, 8059571860456028835, 249785940327179846500, 7837578968934515202570

Offset: 0

Seiichi Manyama, Sep 06 2022

nonn

Cf. A349557, A356785, A356963.

Cf. A356951, A356967.

a(n) = n!*sum(k=0, n\3, (n-k+1)^(k-1)*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));

a(n) = n! * Sum_{k=0..floor(n/3)} (n-k+1)^(k-1) * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).

A356963 E.g.f. satisfies log(A(x)) = x^3/6 * (exp(xA(x)) - 1) A(x).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 2856, 30324, 205920, 1108965, 23266540, 482882686, 6894708184, 75981906455, 1191424314640, 28555511625320, 635370500529216, 11547272842560969, 213976417787571540, 5165550420447656050, 142251473243071866520, 3624660458114853235771

Offset: 0

Seiichi Manyama, Sep 06 2022

nonn

Cf. A349557, A356785, A356962.

Cf. A356952, A356968.

a(n) = n!*sum(k=0, n\4, (n-2*k+1)^(k-1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} (n-2*k+1)^(k-1) * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A371145 E.g.f. satisfies log(A(x)) = x^2A(x)^2 (exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 2550, 20202, 105896, 6501672, 111489930, 1203491630, 53987127612, 1496864088876, 27032265220142, 1088916434686290, 40758246253626960, 1081683296597292752, 44159293393817257746, 1998309768008640244182, 71124972575776526592740

Offset: 0

Seiichi Manyama, Mar 13 2024

nonn

Index entries for reversions of series

Cf. A356785.

a(n) = n!*sum(k=0, n\3, (n+1)^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/3)} (n+1)^(k-1) * Stirling2(n-2*k,k)/(n-2*k)!.

E.g.f.: (1/x) * Series_Reversion( x*exp(x^2*(1 - exp(x))) ). - Seiichi Manyama, Sep 21 2024

A355508 E.g.f. satisfies log(A(x)) = x^2 * (exp(x * A(x)) - 1) * A(x).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 1830, 15162, 82376, 3326472, 59467050, 678585710, 20553790092, 563969783676, 10776243950654, 318310813941330, 10988438698692240, 303144002003606672, 9910024990673571666, 392381835437286982998, 14072003919511407720020

Offset: 0

Seiichi Manyama, Sep 07 2022

nonn

Cf. A349557, A356785, A356892.

Cf. A356962.

m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = Exp[x^2*(Exp[x*A[x]] - 1)*A[x]] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

a(n) = n!*sum(k=0, n\3, (n-k+1)^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/3)} (n-k+1)^(k-1) * Stirling2(n-2*k,k)/(n-2*k)!.

A356882 E.g.f. satisfies: A(x) * log(A(x)) = x * (exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 2, 3, 16, 125, 756, 7567, 85968, 994905, 14373460, 225366251, 3800667960, 72169966453, 1469546796732, 32150706096615, 760806334538656, 19142440567996721, 512272692571487652, 14560087915617858883, 436598686303562722440, 13796641165956117509901

Offset: 0

Seiichi Manyama, Sep 02 2022

nonn

Cf. A349560, A356785, A356788, A356789, A356883.

Cf. A349588.

a(n) = n!*sum(k=0, n\2, (n-2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.

A356883 E.g.f. satisfies: A(x)^2 * log(A(x)) = x * (exp(x*A(x)) - 1).

Original entry on oeis.org

1, 0, 2, 3, -8, 5, 696, 2527, -40144, -178407, 8337880, 76134971, -1781542344, -24938260763, 691630553264, 14216543752335, -312910463346464, -9343318015483471, 195539694928047144, 8145971436703039363, -142317653823753257560, -8498984155838272275459

Offset: 0

Seiichi Manyama, Sep 02 2022

sign

Cf. A349560, A356785, A356788, A356789, A356882.

Cf. A355763.

a(n) = n!*sum(k=0, n\2, (n-3*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} (n-3*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.

cp's OEIS Frontend

A355843 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x).

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A349560 E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * x.

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A356788 E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x)^2.

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A356789 E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x)^3.

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A356962 E.g.f. satisfies log(A(x)) = x^2/2 * (exp(x*A(x)) - 1) * A(x).

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A356963 E.g.f. satisfies log(A(x)) = x^3/6 * (exp(x*A(x)) - 1) * A(x).

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A371145 E.g.f. satisfies log(A(x)) = x^2*A(x)^2 * (exp(x*A(x)) - 1).

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A355508 E.g.f. satisfies log(A(x)) = x^2 * (exp(x * A(x)) - 1) * A(x).

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Mathematica

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A356882 E.g.f. satisfies: A(x) * log(A(x)) = x * (exp(x*A(x)) - 1).

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A349560 E.g.f. satisfies log(A(x)) = (exp(xA(x)) - 1) x.

A356788 E.g.f. satisfies log(A(x)) = x * (exp(xA(x)) - 1) A(x)^2.

A356789 E.g.f. satisfies log(A(x)) = x * (exp(xA(x)) - 1) A(x)^3.

A356962 E.g.f. satisfies log(A(x)) = x^2/2 * (exp(xA(x)) - 1) A(x).

A356963 E.g.f. satisfies log(A(x)) = x^3/6 * (exp(xA(x)) - 1) A(x).

A371145 E.g.f. satisfies log(A(x)) = x^2A(x)^2 (exp(x*A(x)) - 1).