A161633 -id:A161633 - cp's OEIS frontend

A365287 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^3*A(x)^3).

1, 1, 2, 6, 48, 720, 11520, 183960, 3185280, 65681280, 1637193600, 46436544000, 1423113753600, 46607434473600, 1648149184281600, 63369409495392000, 2634451417524326400, 117088187211284889600, 5518546983426135859200, 275022667579200532992000

Offset: 0

Seiichi Manyama, Aug 31 2023

nonn

Cf. A358065, A365285, A365286.

Cf. A161633, A365283.

Join[{1},Table[n!/(n+1) * Sum[(n-3*k)^k * Binomial[n+1,n-3*k]/k!, {k,0,Floor[n/3]}], {n,1,20}]] (* Vaclav Kotesovec, Nov 08 2023 *)

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

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

a(n) ~ 3^(n/3) * (1 + 4*LambertW(3^(1/4)/4))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(3^(1/4)/4)) * 2^(8*n/3 + 4) * exp(n) * LambertW(3^(1/4)/4)^(4*n/3 + 3/2)). - Vaclav Kotesovec, Nov 08 2023

A366235 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(5xA(x)).

Original entry on oeis.org

1, 1, 12, 171, 3644, 104245, 3718470, 159587365, 8014254120, 461209324905, 29936339490050, 2164061360402425, 172443226346717100, 15018744392959920925, 1419463584040707175950, 144700081009666607896125, 15826417814285141247938000, 1848740412846656456007516625

Offset: 0

Paul D. Hanna, Oct 05 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.
In general, if k > 0 and e.g.f. A(x) satisfies A(x) = 1 + x*A(x) * exp(k*x*A(x)), then a(n) ~ k^n * (1 + 2*LambertW(sqrt(k)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(k)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(k)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

			E.g.f.: A(x) = 1 + x + 12*x^2/2! + 171*x^3/3! + 3644*x^4/4! + 104245*x^5/5! + 3718470*x^6/6! + 159587365*x^7/7! + 8014254120*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(5*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+4*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+3*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(+2*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(+1*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-0*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-1*x*A(x))/6! + ...
and
A(x) = 1 + 6*1*6^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 6*2*7^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 6*3*8^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 6*4*9^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 6*5*10^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...

Cf. A161633, A366232, A366233, A366234.

Cf. A365775 (dual).

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(5*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)

/* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 5^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(5*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

a(n) = n! * Sum{k=0..n} binomial(n+1, n-k)/(n+1) * 5^k * (n-k)^k / k!.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 5^k * (n-k)^k/k!.
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x) * exp(5*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(5*x)) ).
(3) A( x/(1 + x*exp(5*x)) ) = 1 + x*exp(5*x).
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-5)*x*A(x)) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-5)*x*A(x)).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-4)*x*A(x)).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
(4.f) A(x) = 1 + 6 * Sum{n>=1} n*(n+5)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
a(n) ~ 5^n * (1 + 2*LambertW(sqrt(5)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(5)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(5)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A371019 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^3*exp(x)) ).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 2280, 35490, 322896, 6532344, 175392720, 3351681630, 74021715240, 2328376978356, 68824597123464, 1989994550546730, 69687384248405280, 2634948077918611440, 98220733842576688416, 3966108617957749165494, 175679596523004500742840

Offset: 0

Seiichi Manyama, Mar 08 2024

nonn

Index entries for reversions of series

Cf. A161633, A371018.

Cf. A365287.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x^3*exp(x)))/x))

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

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

A377554 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^3 ).

Original entry on oeis.org

1, 3, 30, 537, 14124, 493695, 21601458, 1137294039, 70064934600, 4947238170747, 394022075650590, 34951812094581723, 3417754921150904172, 365287875167708973831, 42368411854713294141834, 5300422308901745571018735, 711465905597330333014408848, 101995745742232833085109746803

Offset: 0

Seiichi Manyama, Nov 01 2024

nonn

Index entries for reversions of series

Cf. A161633, A377553.

Cf. A364986.

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

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364986.
a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(3*n+3,k)/(n-k)!.

A364981 E.g.f. satisfies A(x) = 1 + xA(x)exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 4, 39, 580, 11685, 298566, 9248701, 336886936, 14112113049, 668422303210, 35325208755441, 2060811941835780, 131547166492534117, 9120279070776381886, 682489450793082237285, 54828316394224735284016, 4706545644403274325580593

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Cf. A006153, A161633, A364938, A364980.

Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[3*n-2*k+1,k] / ((3*n-2*k+1)*(n-k)!), {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Nov 18 2023 *)

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

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

a(n) ~ sqrt((1 + r*s^3)/(12*s + 9*r*s^4)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.1811100305436879929789759231994897963241226689807... and s = 1.522012903517407628213363540403002787906223513104... are real roots of the system of equations 1 + exp(r*s^3)*r*s = s, 3*r*s^3*(s-1) = 1. - Vaclav Kotesovec, Nov 18 2023

A364982 E.g.f. satisfies A(x) = 1 + xA(x)^2exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 6, 69, 1204, 28345, 842406, 30282385, 1278159240, 61979238513, 3395850105610, 207490382754721, 13989267347891628, 1031687145559176457, 82618837044274734126, 7139807492658000170865, 662286433378726179463696, 65635135687587192429274849

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Cf. A213644, A295238.
Cf. A161633, A364986, A364989.
Cf. A001764.

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

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

A379456 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + x*exp(x)) ).

Original entry on oeis.org

1, 2, 13, 151, 2573, 58221, 1648345, 56138461, 2236816825, 102135829609, 5259937376141, 301678137203433, 19072415186892325, 1317869007328182349, 98818139178323981473, 7991908824553634264101, 693473520767940388417265, 64266613784795934251538513

Offset: 0

Seiichi Manyama, Dec 30 2024

nonn

Cf. A088690, A379846, A379847.
Cf. A108919, A161633.
Cf. A379699, A379700.

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

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

E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(2*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A377551 E.g.f. satisfies A(x) = 1/(1 - x * exp(x*A(x)))^4.

Original entry on oeis.org

1, 4, 28, 348, 6424, 156900, 4788024, 175678468, 7538078944, 370557062532, 20540717542120, 1267858489975044, 86252943572785488, 6412719306748404676, 517341051818833834648, 45012757582472804739780, 4201834386001491870902464, 418891045572216881436564228

Offset: 0

Seiichi Manyama, Oct 31 2024

nonn

Cf. A161633, A377541, A377545.

Cf. A377550, A377552.

a(n) = 4*n!*sum(k=0, n, k^(n-k)*binomial(4*n-3*k+4, k)/((4*n-3*k+4)*(n-k)!));

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377550.

a(n) = 4 * n! * Sum_{k=0..n} k^(n-k) * binomial(4*n-3*k+4,k)/( (4*n-3*k+4)*(n-k)! ).

A365175 E.g.f. satisfies A(x) = 1 + xA(x)^4exp(x*A(x)).

Original entry on oeis.org

1, 1, 10, 189, 5476, 215145, 10701006, 644909503, 45687408712, 3721382812305, 342689189598010, 35206864089944151, 3992473080042706524, 495361299387667990537, 66752437447119717428422, 9708649781691227748131535, 1515863453268825963300368656

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Cf. A364987, A364989, A365176, A365177.

Cf. A161633, A213644, A364984.

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

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

A371018 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^2*exp(x)) ).

Original entry on oeis.org

1, 0, 2, 6, 60, 620, 7950, 129402, 2365496, 50512968, 1208642490, 32223422990, 947694971652, 30435132773916, 1061061668979494, 39889366397571810, 1608910488000292080, 69305890226183224592, 3175519952912430375666, 154216789672147809137046

Offset: 0

Seiichi Manyama, Mar 08 2024

nonn

Index entries for reversions of series

Cf. A161633, A371019.

Cf. A365283.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x^2*exp(x)))/x))

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

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

cp's OEIS Frontend

A365287 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^3*A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A366235 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(5*x*A(x)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A371019 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^3*exp(x)) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A377554 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^3 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A364981 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A364982 E.g.f. satisfies A(x) = 1 + x*A(x)^2*exp(x*A(x)^2).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A379456 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + x*exp(x)) ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A377551 E.g.f. satisfies A(x) = 1/(1 - x * exp(x*A(x)))^4.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365175 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)).

Views

Author

Keywords

Crossrefs

Programs

A365287 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^3*A(x)^3).

A366235 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(5xA(x)).

A364981 E.g.f. satisfies A(x) = 1 + xA(x)exp(x*A(x)^3).

A364982 E.g.f. satisfies A(x) = 1 + xA(x)^2exp(x*A(x)^2).

A365175 E.g.f. satisfies A(x) = 1 + xA(x)^4exp(x*A(x)).