A184949 -id:A184949 - 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

A355842 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)).

Original entry on oeis.org

1, 0, 2, 3, 44, 210, 3054, 27300, 449952, 6020784, 115381080, 2053568880, 45733246536, 1010390340960, 25916586868704, 680621684914080, 19881379012231680, 603034125051738240, 19833651290982164544, 680927283288289169280, 24953207662252739030400

Offset: 0

Seiichi Manyama, Jul 18 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A052813, A184949, A355779.

nmax = 20; CoefficientList[Series[LambertW[x * Log[1-x]] / (x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 21 2022 *)

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

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

E.g.f.: exp( -LambertW(x * log(1-x)) ).
E.g.f.: LambertW(x * log(1-x)) / (x * log(1-x)).
a(n) ~ sqrt(1 + exp(1)*r^2/(1-r)) * n^(n-1) / (exp(n-1) * r^n), where r = 0.5123112855238643734867005914814802444318611742265... is the positive root of the equation r*log(1-r) = -exp(-1). - Vaclav Kotesovec, Jul 21 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Aug 28 2022

A349559 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^x.

Original entry on oeis.org

1, 0, 2, 3, 44, 270, 3714, 44940, 746528, 13149864, 271954440, 6154715160, 155055594792, 4254730262640, 127019898548256, 4088313657038520, 141301521555548160, 5216698433745207360, 204946906542573645504, 8536144551987171202560

Offset: 0

Seiichi Manyama, Nov 22 2021

nonn

Cf. A007113, A349556.

Cf. A184949, A356786, A356787.

a:= n-> n!*coeff(series(RootOf(1/(1-x*A)^x-A, 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] = 1/(1 - x*A[x])^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=(1-x*A)^(-x+x*O(x^n))); Vec(serlaplace(A))

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

a(n) ~ sqrt(1 + r - 2*r*log(r)) * n^(n-1) / ((1+r)^2 * exp(n) * r^(n + 1/2)), where r = 0.4214518303433019663622598075106479936652984008256... is the root of the equation r^(1-r) * (1+r)^(1+r) = 1. - Vaclav Kotesovec, Nov 22 2021

a(n) = n! * Sum_{k=0..floor(n/2)} (n-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Aug 27 2022

A356786 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x * A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 92, 510, 15114, 174300, 5558944, 103712616, 3672530280, 96397602840, 3830335035240, 129817630491120, 5796134828193696, 239906921239210680, 11996259216566469120, 584024600798956215360, 32523678395272329425856

Offset: 0

Seiichi Manyama, Aug 27 2022

nonn

Cf. A184949, A349559, A356787.

Cf. A355766.

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

a(n) = n! * Sum_{k=0..floor(n/2)} (n+k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

A356787 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x * A(x)^3).

Original entry on oeis.org

1, 0, 2, 3, 116, 630, 24054, 273000, 11105072, 207213552, 9175467960, 245785969440, 11954556125544, 421832039016960, 22609694372667024, 991695134898861120, 58565049582761702400, 3065736317041568378880, 199024242549235933723200

Offset: 0

Seiichi Manyama, Aug 27 2022

nonn

Cf. A184949, A349559, A356786.

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

a(n) = n! * Sum_{k=0..floor(n/2)} (n+2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

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

Seiichi Manyama, Sep 06 2022

nonn

Cf. A184949, A349556, A356968.

Cf. A356752, A356962.

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)!));

a(n) = n! * Sum_{k=0..floor(n/3)} (n-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).

A356968 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^3/6 * A(x)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 4144, 40320, 409800, 4527600, 72552480, 1170449280, 19489513680, 338983444800, 6672681818880, 141166715289600, 3149324442700800, 73497460049395200, 1825098639493104000, 47984287767342796800, 1326460667797094860800

Offset: 0

Seiichi Manyama, Sep 06 2022

nonn

Cf. A184949, A349556, A356967.

Cf. A356753, A356963.

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

a(n) = n! * Sum_{k=0..floor(n/4)} (n-2*k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).

A356884 E.g.f. satisfies A(x)^A(x) = 1/(1 - x*A(x))^x.

Original entry on oeis.org

1, 0, 2, 3, 20, 150, 1254, 14280, 190000, 2863728, 49465080, 954312480, 20303200488, 473604468480, 12007399511184, 328671680500800, 9663415159357440, 303695188102656000, 10159173955921651776, 360424299614544829440, 13517056067747847719040

Offset: 0

Seiichi Manyama, Sep 02 2022

nonn

Cf. A184949, A349559, A356786, A356787, A356885.

Cf. A141209.

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

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

A356885 E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x*A(x))^x.

Original entry on oeis.org

1, 0, 2, 3, -4, 30, 954, 6300, 6432, 424872, 18273960, 260682840, 1754408424, 47063118960, 2314149100704, 54798086299320, 773632032345600, 20746972036284480, 1072205580591921600, 36098491880448944640, 816375193722964932480, 25160238159364392336000

Offset: 0

Seiichi Manyama, Sep 02 2022

sign

Cf. A184949, A349559, A356786, A356787, A356884.

Cf. A355767.

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

a(n) = n! * Sum_{k=0..floor(n/2)} (n-3*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

A371147 E.g.f. satisfies A(x) = 1/(1 - xA(x))^(x^2A(x)^2).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 2700, 21168, 173040, 7611840, 125415360, 1848591360, 71211998880, 1844075600640, 41500836336960, 1609925690419200, 55708371912211200, 1719475797882286080, 72091120665964654080, 3046645011228998860800, 119757727692451283097600

Offset: 0

Seiichi Manyama, Mar 13 2024

nonn

Index entries for reversions of series

Cf. A184949, A371145.

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

a(n) = n! * Sum_{k=0..floor(n/3)} (n+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.

E.g.f.: (1/x) * Series_Reversion( x*(1 - x)^(x^2) ). - Seiichi Manyama, Sep 21 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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A355842 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)).

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A349559 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^x.

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A356786 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x * A(x)^2).

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A356787 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x * A(x)^3).

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A356967 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^2/2 * A(x)).

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A356968 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x^3/6 * A(x)).

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A356884 E.g.f. satisfies A(x)^A(x) = 1/(1 - x*A(x))^x.

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PARI

Formula

A356885 E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x*A(x))^x.

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Author

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Crossrefs

Programs

PARI

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

A371147 E.g.f. satisfies A(x) = 1/(1 - xA(x))^(x^2A(x)^2).