A349557 -id:A349557 - cp's OEIS frontend

A356785 E.g.f. satisfies log(A(x)) = x * (exp(xA(x)) - 1) A(x).

1, 0, 2, 3, 64, 365, 7356, 85687, 1920752, 34821369, 905128300, 22172123171, 672107454888, 20552960420005, 721088019634724, 26257726364294895, 1053711696230404576, 44336326818388565105, 2010106841636689325532, 95747319823049127621019

Offset: 0

Seiichi Manyama, Aug 27 2022

nonn

Index entries for reversions of series

Cf. A349560, A356788, A356789.

Cf. A184949, A349557, A355843.

nmax = 19; A[_] = 1;
Do[A[x_] = Exp[x*(Exp[x*A[x]]-1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

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

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

A349588 E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)) - 1.

Original entry on oeis.org

1, 1, 2, 8, 47, 367, 3592, 42317, 583522, 9223872, 164482761, 3267077365, 71540314562, 1712334954865, 44479256704898, 1246241906483516, 37465750470667023, 1202986323660907447, 41089436549405467096, 1487622596267089224901, 56907111260864275384346

Offset: 0

Seiichi Manyama, Nov 22 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..403

Cf. A141209, A349587, A349589.

Cf. A349557.

a[n_] := Sum[(n - k + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 23 2021 *)

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

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

a(n) ~ s * n^(n-1) / (sqrt(r*s - 1/(1 + log(s))) * exp(n) * r^n), where r = 0.4858893246242883887847088396703818017675758048583... and s = 3.016426175038226058288579473351450432292607021364... are roots of the system of equations exp(r*s) = 1 + s*log(s), exp(r*s)*r = 1 + log(s). - Vaclav Kotesovec, Nov 25 2021

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

A349558 E.g.f. satisfies: log(A(x)) = (1 - exp(-xA(x))) A(x).

Original entry on oeis.org

1, 1, 4, 32, 393, 6547, 138046, 3525853, 105832964, 3651748332, 142429413387, 6196895235709, 297571887174040, 15632879134292045, 891910713837242092, 54919409605089141532, 3630105859259972654905, 256374187841461047791587

Offset: 0

Seiichi Manyama, Nov 21 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..360

Cf. A058864, A349557.

a[n_] := Sum[(-1)^(n - k) * (n + k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 22 2021 *)

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

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

a(n) ~ sqrt((s-1)*s^3 / (1 + r*(2*s - 3)*s - r^2*(s-1)*s^2)) * n^(n-1) / (exp(n) * r^(n -1/2)), where r = 0.2202409288542107090687589144963703329896230236509... and s = 1.7315644042495989781932730410872588555151921253414... are roots of the system of equations s = s/exp(r*s) + log(s), (s-1)/s - (1 - r*s)/exp(r*s) = 0. - Vaclav Kotesovec, Nov 22 2021

A355762 E.g.f. satisfies log(A(x)) = (exp(xA(x)) - 1) A(x)^2.

Original entry on oeis.org

1, 1, 8, 125, 2987, 96727, 3963841, 196769897, 11480304448, 770031502467, 58386951857583, 4938864464154469, 461111056016847137, 47101341445053180079, 5225323162578044669492, 625646891309723527419137, 80416734865584980392853799, 11044230667889978466327860347

Offset: 0

Seiichi Manyama, Jul 16 2022

nonn

Cf. A030019, A349557.

Cf. A162656, A355763, A355766.

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

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

A356973 E.g.f. satisfies log(A(x)) = (exp(x * A(x)^3) - 1) * A(x).

Original entry on oeis.org

1, 1, 10, 206, 6555, 283777, 15577332, 1036984027, 81191314678, 7311591070938, 744577308572189, 84608911909469235, 10613728203840498210, 1456899252646375490851, 217215453964895439271178, 34956361099228031471844962, 6039398076840098381458042875

Offset: 0

Seiichi Manyama, Sep 07 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..319

Cf. A052880, A349557, A356972.

Cf. A216136, A356960.

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

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

A356972 E.g.f. satisfies log(A(x)) = (exp(x * A(x)^2) - 1) * A(x).

Original entry on oeis.org

1, 1, 8, 128, 3139, 104382, 4393590, 224045271, 13428576766, 925335827928, 72082558060889, 6264277731652096, 600873473776204782, 63059026039778220285, 7187299097301622432156, 884141943373486896560252, 116756337165196381259759707, 16474480747756013055963484442

Offset: 0

Seiichi Manyama, Sep 07 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..329

Cf. A052880, A349557, A356973.

Cf. A216135, A349601, A356914.

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

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

A357346 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(A(x)).

Original entry on oeis.org

0, 1, 5, 52, 849, 18996, 540986, 18726247, 763480675, 35837071558, 1903538106065, 112880374866172, 7392418912962210, 529898419942327801, 41266682731537698181, 3469461853041348996044, 313200848521114144611273, 30215925892728362737156556

Offset: 0

Seiichi Manyama, Sep 25 2022

nonn

Index entries for reversions of series

Cf. A052888, A357347, A357348, A357424.

Cf. A048802, A349557.

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

a(n) = Sum_{k=1..n} (n+k)^(k-1) * Stirling2(n,k).

E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(-x)) ). - Seiichi Manyama, Sep 09 2024

cp's OEIS Frontend

A356785 E.g.f. satisfies log(A(x)) = x * (exp(x*A(x)) - 1) * A(x).

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A349588 E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)) - 1.

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

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A349558 E.g.f. satisfies: log(A(x)) = (1 - exp(-x*A(x))) * A(x).

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A355762 E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * A(x)^2.

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A356973 E.g.f. satisfies log(A(x)) = (exp(x * A(x)^3) - 1) * A(x).

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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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A356972 E.g.f. satisfies log(A(x)) = (exp(x * A(x)^2) - 1) * A(x).

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A356785 E.g.f. satisfies log(A(x)) = x * (exp(xA(x)) - 1) A(x).

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

A349558 E.g.f. satisfies: log(A(x)) = (1 - exp(-xA(x))) A(x).

A355762 E.g.f. satisfies log(A(x)) = (exp(xA(x)) - 1) A(x)^2.

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).