A161633 -id:A161633 - cp's OEIS frontend

A213644 E.g.f. satisfies: A(x) = 1 + xA(x)^2exp(x*A(x)).

1, 1, 6, 63, 988, 20725, 546246, 17364445, 646910328, 27652214313, 1334291800330, 71749167806041, 4255000637001588, 275904038948566093, 19420072633921942542, 1474700254793433800805, 120174737260376219862256, 10461031446553525766071249, 968785652772129485926955538

Offset: 0

Paul D. Hanna, Jun 17 2012

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...
such that A(x-x^2*exp(x)) = 1/(1-x*exp(x)) where:
1/(1-x*exp(x)) = 1 + x + 4*x^2/2! + 21*x^3/3! + 148*x^4/4! + 1305*x^5/5! +...+ A006153(n)*x^n/n! +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A213643, A006153, A161633.

Flatten[{1,Table[1/(n+1)*Sum[k^(n-k)/(n-k)!*(n+k)!/k!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 15 2013 *)

{a(n)=1/(n+1)*sum(k=0, n, k^(n-k)/(n-k)! * (n+k)!/k! )}

{a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*exp(x+x*O(x^n))),n)}
for(n=0,25,print1(a(n),", "))

E.g.f. satisfies: A(x) = 1/(1 - x*A(x)*exp(x*A(x))).
E.g.f. satisfies: A(x-x^2*exp(x)) = 1/(1-x*exp(x)).
a(n) = 1/(n+1) * Sum_{k=0..n} k^(n-k)/(n-k)! * (n+k)!/k!.
a(n) = A213643(n+1)/(n+1).
Limit n->infinity (a(n)/n!)^(1/n) = r*(1+r)/(1-r) = 5.5854662015218413..., where r = 0.7603592340333989... is the root of the equation (1-r^2)/r^2 = exp((r-1)/r), same as for A213643. - Vaclav Kotesovec, Jul 15 2013
a(n) ~ (1+r)*sqrt(r/(1+2*r-r^2)) * n^(n-1) * (r*(1+r)/(1-r))^n / exp(n). - Vaclav Kotesovec, Dec 28 2013

A161630 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

Original entry on oeis.org

1, 1, 3, 19, 181, 2321, 37501, 731935, 16758393, 440525377, 13077834841, 432796650551, 15799794395749, 630773263606513, 27339525297079269, 1278550150117141231, 64171287394646697841, 3440711053857464325377

Offset: 0

Paul D. Hanna, Jun 17 2009

nonn

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 + x^4*A(x)^4 +...

G. C. Greubel, Table of n, a(n) for n = 0..373
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265

Cf. A161633 (e.g.f. = log(A(x))/x).

Cf. A212722, A212917, A245265, A125500.

Table[Sum[n! * (n-k+1)^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(n-1,n-k)))}

{a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(x/(1-x*A)));n!*polcoeff(A^m,n)}

a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(n-1,n-k).
E.g.f. satisfies: A(x) = exp(x) * A(x)^(x*A(x)). - Paul D. Hanna, Aug 02 2013
a(n) ~ n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2) * exp(n) * c^(2*n+3/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

A366232 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(2xA(x)).

Original entry on oeis.org

1, 1, 6, 54, 728, 13000, 290352, 7800016, 245115264, 8826560640, 358463525120, 16212238054144, 808215885708288, 44035925223746560, 2603618739995621376, 166031767704180111360, 11359670347331723952128, 830065763154102656204800, 64518486557995327748898816

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.

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 54*x^3/3! + 728*x^4/4! + 13000*x^5/5! + 290352*x^6/6! + 7800016*x^7/7! + 245115264*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(2*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+1*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(-0*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(-1*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-2*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-3*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-4*x*A(x))/6! + ...
and
A(x) = 1 + 3*1*3^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 3*2*4^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 3*3*5^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 3*4*6^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 3*5*7^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ... + A212722(n)*x^n/n! + ...

Cf. A161633, A366233, A366234, A366235.

Cf. A365772 (dual), A212722 (exp(x*A(x))).

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(2*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) * 2^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(2*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) * 2^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) * 2^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(2*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(2*x)) ).
(3) A( x/(1 + x*exp(2*x)) ) = 1 + x*exp(2*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-2)*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-2)*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-1)*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*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+1)*x*A(x)).
a(n) ~ n^(n-1) * (1 + 2*LambertW(1/sqrt(2)))^(n + 3/2) / (sqrt(1 + LambertW(1/sqrt(2))) * 2^(n+2) * exp(n) * LambertW(1/sqrt(2))^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A366233 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(3xA(x)).

Original entry on oeis.org

1, 1, 8, 87, 1428, 31125, 847818, 27785205, 1065267864, 46802921769, 2319200977230, 127985909702409, 7785440359004916, 517616528753919933, 37344834374921549154, 2906043724955696034285, 242627026212699695954352, 21634774261037677172406609, 2052077586846349144929739542

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.

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 87*x^3/3! + 1428*x^4/4! + 31125*x^5/5! + 847818*x^6/6! + 27785205*x^7/7! + 1065267864*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(3*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+2*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+1*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(-0*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(-2*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-3*x*A(x))/6! + ...
and
A(x) = 1 + 4*1*4^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 4*2*5^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 4*3*6^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 4*4*7^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 4*5*8^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + 254701*x^6/6! + ... + A212917(n)*x^n/n! + ...

Cf. A161633, A366232, A366234, A366235.

Cf. A365773 (dual), A212917 (exp(x*A(x))).

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(3*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) * 3^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(3*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) * 3^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) * 3^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(3*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(3*x)) ).
(3) A( x/(1 + x*exp(3*x)) ) = 1 + x*exp(3*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-3)*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-3)*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-2)*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-1)*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*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)).
a(n) ~ 3^n * (1 + 2*LambertW(sqrt(3)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(3)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(3)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A364980 E.g.f. satisfies A(x) = 1 + xA(x)exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 33, 412, 6945, 147846, 3807601, 115151464, 4001162913, 157096369450, 6878742553881, 332361857826780, 17566215943990753, 1008161606338206334, 62440146891413434305, 4151012174991960338896, 294834882756167048975553

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Cf. A006153, A161633, A161635, A364981.

Join[{1}, Table[n! * Sum[k^(n-k) * Binomial[2*n-k+1,k] / ((2*n-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(2*n-k+1, k)/((2*n-k+1)*(n-k)!));

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

a(n) ~ sqrt((1 + r*s^2)/(6 + 4*r*s^2)) * n^(n-1) / (exp(n) * r^(n + 1/2)), where r = 0.2190923703746024362724546703711998154573791458000... and s = 1.747404632046819382844696016554403302840973484745... are real roots of the system of equations 1 + exp(r*s^2)*r*s = s, 2*r*s^2*(s-1) = 1. - Vaclav Kotesovec, Nov 18 2023

A364989 E.g.f. satisfies A(x) = 1 + xA(x)^4exp(x*A(x)^4).

Original entry on oeis.org

1, 1, 10, 207, 6628, 288885, 15969606, 1070760523, 84448152328, 7660906993737, 785932068816010, 89973000854464431, 11370915080258640204, 1572520778920744136029, 236212754707591898128270, 38299196311415039667233715, 6666717272317556205911393296

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Cf. A161633, A364982, A364986.

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*n+1, k)/(n-k)!)/(4*n+1);

a(n) = (n!/(4*n+1)) * Sum_{k=0..n} k^(n-k) * binomial(4*n+1,k)/(n-k)!.

A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(4xA(x)).

Original entry on oeis.org

1, 1, 10, 126, 2392, 60600, 1916304, 72917488, 3246171520, 165609099648, 9529240349440, 610657739172096, 43136025287678976, 3330356645773880320, 279024535906794539008, 25214258236430338160640, 2444656672390982922502144, 253144081975231633923342336

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.

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 126*x^3/3! + 2392*x^4/4! + 60600*x^5/5! + 1916304*x^6/6! + 72917488*x^7/7! + 3246171520*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(4*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+3*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+2*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(+1*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-0*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-1*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-2*x*A(x))/6! + ...
and
A(x) = 1 + 5*1*5^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 5*2*6^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 5*3*7^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 5*4*8^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 5*5*9^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ... + A245265(n)*x^n/n! + ...

Cf. A161633, A366232, A366233, A366235.

Cf. A365774 (dual), A245265 (exp(x*A(x))).

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(4*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) * 4^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(4*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) * 4^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) * 4^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(4*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(4*x)) ).
(3) A( x/(1 + x*exp(4*x)) ) = 1 + x*exp(4*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-4)*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-4)*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-3)*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-2)*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-1)*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*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+1)*x*A(x)).
a(n) ~ (1 + 2*LambertW(1))^(n + 3/2) * n^(n-1) / (4 * sqrt(1 + LambertW(1)) * exp(n) * LambertW(1)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A377553 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^2 ).

Original entry on oeis.org

1, 2, 14, 174, 3176, 77010, 2336892, 85316714, 3644408336, 178412603778, 9851421767060, 605826315779322, 41068369222584024, 3042849619010389058, 244657525386435161756, 21217387476442659806250, 1974219906922046702054432, 196191093901292764305110274, 20739322455031604846405387556

Offset: 0

Seiichi Manyama, Nov 01 2024

nonn

Index entries for reversions of series

Cf. A161633, A377554.

Cf. A364982.

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

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364982.
a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(2*n+2,k)/(n-k)!.

A364986 E.g.f. satisfies A(x) = 1 + xA(x)^3exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 8, 129, 3196, 107465, 4575966, 236120059, 14322901832, 998966928897, 78770826493210, 6929685905371691, 672900446143476156, 71491442785783506577, 8249400210035835040022, 1027394346436911560475915, 137360293432089585554830096

Offset: 0

Seiichi Manyama, Aug 15 2023

nonn

Cf. A364983, A364984, A364985.
Cf. A161633, A364982, A364989.
Cf. A002293.

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

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

A365283 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^2*A(x)^2).

Original entry on oeis.org

1, 1, 2, 12, 120, 1380, 19440, 341040, 7029120, 164762640, 4355769600, 128527439040, 4181332700160, 148633442717760, 5734427199621120, 238676208285715200, 10659325532663808000, 508452777299622355200, 25800664274991135129600

Offset: 0

Seiichi Manyama, Aug 31 2023

nonn

Cf. A358064, A365282, A365284.

Cf. A161633, A365287.

Join[{1},Table[n!/(n+1) * Sum[(n-2*k)^k * Binomial[n+1,n-2*k]/k!, {k, 0, Floor[n/2]}], {n,1,20}]] (* Vaclav Kotesovec, Nov 08 2023 *)

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

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

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

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A213644 E.g.f. satisfies: A(x) = 1 + x*A(x)^2*exp(x*A(x)).

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A161630 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

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A366232 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(2*x*A(x)).

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A366233 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(3*x*A(x)).

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A364980 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x*A(x)^2).

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A364989 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)^4).

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A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(4*x*A(x)).

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A377553 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^2 ).

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A213644 E.g.f. satisfies: A(x) = 1 + xA(x)^2exp(x*A(x)).

A366232 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(2xA(x)).

A366233 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(3xA(x)).

A364980 E.g.f. satisfies A(x) = 1 + xA(x)exp(x*A(x)^2).

A364989 E.g.f. satisfies A(x) = 1 + xA(x)^4exp(x*A(x)^4).

A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(4xA(x)).

A364986 E.g.f. satisfies A(x) = 1 + xA(x)^3exp(x*A(x)^3).

A365283 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^2*A(x)^2).