A213643 -id:A213643 - 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

A161629 E.g.f. satisfies: A(x) = exp( xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2x) is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 25, 349, 6821, 171421, 5265625, 191160201, 8007548617, 380157603481, 20171371753061, 1182973489103869, 75984447924612397, 5305029326492409333, 400014338565211619761, 32396515980658185762961

Offset: 0

Paul D. Hanna, Jun 17 2009

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 14*x^4*A(x)^4 +...+ A000108(n)*x^n*A(x)^n +...
log(A(x))/x = 1 + x + 6*x^2/2! + 63*x^3/3! + 988*x^4/4! + 20725*x^5/5! +...+ A213644(n)*x^n/n! +...
log(A(x)) = x + 2*x^2/2! + 18*x^3/3! + 252*x^4/4! + 4940*x^5/5! +...+ A213643(n)*x^n/n! +...
Ordinary Generating Function:
O.g.f.: 1 + x + 3*x^2 + 25*x^3 + 349*x^4 + 6821*x^5 + 171421*x^6 +...
O.g.f.: 1 + x/(1-x) + 2*x^2/(1-2*x)^3 + 6*2!*x^3/(1-3*x)^5 + 20*3!*x^4/(1-4*x)^7 + 70*4!*x^5/(1-5*x)^9 + 252*5!*x^6/(1-6*x)^11 +...+ (2*n-2)!/(n-1)!*x^n/(1-n*x)^(2*n-1) +...

Cf. A213643 (log), A214689, A000108.

Flatten[{1,Table[Sum[n!/k!*(n-k+1)^(k-1)*Binomial[2*n-k, n-k]*k/(2*n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 26 2014 *)

{a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(m+n-k)^(k-1)*binomial(2*n-k,n-k)*k/(2*n-k)))}

{a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp((1-sqrt(1-4*x*A))/(2*A)));n!*polcoeff(A^m,n)}

/* O.g.f.: */
{a(n)=polcoeff(1+sum(m=1, n, (2*m-2)!/(m-1)!*x^m/(1-m*x+x*O(x^n))^(2*m-1)), n)}

E.g.f.: A(x) = exp(F(x)) where F(x) = x + F(x)^2*exp(F(x)) is the e.g.f. of A213643.
E.g.f.: A(x) = Sum_{n>=0} a(n)*x^n/n!, where
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(2*n-k,n-k)*k/(2*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(2*n-k,n-k)*k/(2*n-k).
...
O.g.f.: A(x) = 1 + Sum_{n>=1} (2*n-2)!/(n-1)! * x^n/(1 - n*x)^(2*n-1).
a(n) ~ n^(n-1) * sqrt((r*s^3*(1-6*r*s+8*r^2*s^2)) / (1 - (-2+8*r+r^2)*s + 4*r*(-4+4*r+r^2)*s^2 + 4*r^2*(8+r)*s^3)) / (exp(n) * r^n), where s = 1.370489293947401403417767032... is the root of the equation log(s)*(1-s*log(s)) + 2*(1+s) = (1+2*s) * sqrt((1+s)/s), and r = log(s)*(1-s*log(s)) = 0.179036084709909351719214... - Vaclav Kotesovec, Feb 26 2014

A143136 E.g.f. satisfies: A(x) = x + sinh( A(x) )^2.

Original entry on oeis.org

1, 2, 12, 128, 1920, 36992, 870912, 24232448, 777999360, 28309164032, 1151292628992, 51750540443648, 2547747292446720, 136336755956252672, 7879446478581399552, 489119124160488931328, 32456290094449950720000

Offset: 1

Paul D. Hanna, Jul 27 2008

nonn

Radius of convergence is r = log(sqrt(2)+1)/2 - (sqrt(2)-1)/2 = 0.2335800...,

where A(r) = log(1+sqrt(2))/2 = arcsinh(1)/2 = 0.44068679...

			A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ...
sinh(A(x)) = G(x) is the e.g.f. of A143137:
G(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2101*x^5/5! + ...
Related expansions:
A(x) = x + sinh(x)^2 + d/dx sinh(x)^4/2! + d^2/dx^2 sinh(x)^6/3! + d^3/dx^3 sinh(x)^8/4! + ...
log(A(x)/x) = sinh(x)^2/x + d/dx (sinh(x)^4/x)/2! + d^2/dx^2 (sinh(x)^6/x)/3! + d^3/dx^3 (sinh(x)^8/x)/4! +...

Alois P. Heinz, Table of n, a(n) for n = 1..150

Cf. A143134, A143137, A213643.

Rest[CoefficientList[InverseSeries[Series[x - Sinh[x]^2,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)

{a(n)=n!*polcoeff(serreverse(x-sinh(x+x*O(x^n))^2),n)}

{a(n)=local(A=x);for(i=0,n,A=x + sinh(A)^2);n!*polcoeff(A,n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, sinh(x+x*O(x^n))^(2*m)/m!)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, sinh(x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

E.g.f.: A(x) = Series_Reversion( x - sinh(x)^2 ).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x)^(2*n)/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (sinh(x)^(2*n)/x)/n! ).
E.g.f. derivative: A'(x) = 1/(1 - sinh(2*A(x))).
a(n) ~ 2^(n-5/4) * n^(n-1) / (exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014

A214688 E.g.f. equals the series reversion of x - x^2exp(2x).

Original entry on oeis.org

1, 2, 24, 408, 9760, 299520, 11223744, 496802432, 25365482496, 1467480983040, 94873742909440, 6778628603670528, 530412734126346240, 45110083291805622272, 4143219058165730672640, 408715543077297795072000, 43097868598208296895512576, 4837629293480336802779234304

Offset: 1

Paul D. Hanna, Aug 01 2012

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 24*x^3/3! + 408*x^4/4! + 9760*x^5/5! + ...
where A(x - x^2*exp(2*x)) = x and A(x) = x + A(x)^2*exp(2*A(x)).
The e.g.f. satisfies:
(3) A(x) = x + x^2*exp(2*A(x)) + 2*x^3*exp(4*A(x)) + 5*x^4*exp(6*A(x)) + 14*x^5*exp(8*A(x)) + 42*x^6*exp(10*A(x)) + ...
(4) log(A(x)/x) = x*exp(2*A(x)) + 3*x^2*exp(4*A(x))/2 + 10*x^3*exp(6*A(x))/3 + 35*x^4*exp(8*A(x))/4 + 126*x^5*exp(10*A(x))/5 + ...
(5) A(x) = x + x^2*exp(2*x) + d/dx x^4*exp(4*x)/2! + d^2/dx^2 x^6*exp(6*x)/3! + d^3/dx^3 x^8*exp(8*x)/4! + ...
(6) log(A(x)/x) = x*exp(2*x) + d/dx x^3*exp(4*x)/2! + d^2/dx^2 x^5*exp(6*x)/3! + d^3/dx^3 x^7*exp(8*x)/4! + ...
Related expansions:
A(x) = x*Catalan(x*G(x)) where G(x) = exp(2*A(x)):
exp(A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 529*x^4/4! + 12601*x^5/5! + 385891*x^6/6! + ...
exp(2*A(x)) = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1360*x^4/4! + 32352*x^5/5! + 989824*x^6/6! + ..., which is the e.g.f. of A214689.
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 216*x^4/4! + 5040*x^5/5! + 153120*x^6/6! + ...
Ordinary Generating Function:
O.g.f.: x + 2*x^2 + 24*x^3 + 408*x^4 + 9760*x^5 + 299520*x^6 + ...
O.g.f.: x + 2*x^2/(1-2*x)^3 + 6*2!*x^3/(1-4*x)^5 + 20*3!*x^4/(1-6*x)^7 + 70*4!*x^5/(1-8*x)^9 + 252*5!*x^6/(1-10*x)^11 + ... + (2*n)!/n!*x^(n+1)/(1-2*n*x)^(2*n+1) + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A213643, A214689, A215003.

A214688:= func< n | n eq 1 select 1 else (&+[Binomial(n-1,k)*Binomial(n+k-1,k)*Factorial(k)*(2*k)^(n-k-1): k in [1..n-1]]) >;
[A214688(n): n in [1..30]]; // G. C. Greubel, Mar 07 2024

Flatten[{1,Table[Sum[(2*k)^(n-k-1)/(n-k-1)!*(n+k-1)!/k!,{k,1,n-1}],{n,2,20}]}] (* Vaclav Kotesovec, Jul 13 2013 *)

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

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

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, exp(2*m*x+x*O(x^n))*x^(2*m)/m!)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, exp(2*m*x+x*O(x^n))*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

/* From o.g.f.: */
{a(n)=polcoeff(sum(m=0, n, (2*m)!/m!*x^(m+1)/(1-2*m*x+x*O(x^n))^(2*m+1)), n)}

def A214688(n): return int(n==1)+sum(binomial(n-1,k)*binomial(n+k-1,k)*factorial(k)*(2*k)^(n-k-1) for k in range(1,n))
[A214688(n) for n in range(1,30)] # G. C. Greubel, Mar 07 2024

a(n) = Sum_{k=0..n-1} ( (n+k-1)! * (2*k)^(n-k-1) )/(k! * (n-k-1)!).
O.g.f.: Sum_{n>=0} (2*n)!/n! * x^(n+1) / (1 - 2*n*x)^(2*n+1).
E.g.f. satisfies:
(1) A(x) = x + A(x)^2*exp(2*A(x)).
(2) A(x) = x*Catalan( x*exp(2*A(x)) ) where Catalan(x) = (1-sqrt(1-4*x))/(2*x).
(3) A(x) = x*Sum_{n>=0} binomial(2*n+1,n)/(2*n+1) * x^n * exp(2*n*A(x)).
(4) A(x) = x*exp( Sum_{n>=1} binomial(2*n-1,n) * x^n * exp(2*n*A(x)) / n ).
(5) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * exp(2*n*x) / n!.
(6) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * exp(2*n*x) / n! ).
(7) A(x) = log(G(x))/2 where G(x) = exp(2*x*Catalan(x*G(x))) is the e.g.f. of A214689, and Catalan(x) = (1-sqrt(1-4*x))/(2*x).
Limit_{n->oo} (a(n)/n!)^(1/n) = (2*r*(1+r))/(1-r) = 6.801725926701655517492664481..., where r = 0.670589533381613711... is the root of the equation (1-r^2)/(2*r^2) = exp((r-1)/r). - Vaclav Kotesovec, Jul 13 2013
a(n) ~ (1-r) * n^(n-1) * (2*r*(1+r)/(1-r))^n / (2*sqrt(r*(1+2*r-r^2)) * exp(n)). - Vaclav Kotesovec, Dec 28 2013

Name changed and entry revised by Paul D. Hanna, Jul 13 2013

A215003 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x))/2.

Original entry on oeis.org

1, 1, 6, 51, 610, 9360, 175371, 3881269, 99083916, 2866173795, 92650139560, 3309877247886, 129495296417565, 5506601964331741, 252882022593123210, 12473008516763238375, 657621285983402967766, 36908182475893682882532, 2196924777991866630353571

Offset: 1

Paul D. Hanna, Jul 30 2012

nonn

			E.g.f.: A(x) = x + x^2/2! + 6*x^3/3! + 51*x^4/4! + 610*x^5/5! + 9360*x^6/6! +...
where A(x - x^2*exp(x)/2) = x and A(x) = x + A(x)^2*exp(A(x))/2.
Related expansions:
A(x)^2 = 2*x^2/2! + 6*x^3/3! + 54*x^4/4! + 630*x^5/5! + 9570*x^6/6! +...
exp(A(x)) = 1 + x + 2*x^2/2! + 10*x^3/3! + 85*x^4/4! + 1011*x^5/5! + 15466*x^6/6! + 289213*x^7/7! +...
A(x) = x + exp(x)*x^2/2 + d/dx exp(2*x)*x^4/2!/4 + d^2/dx^2 exp(3*x)*x^6/3!/8 + d^3/dx^3 exp(4*x)*x^8/4!/16 +...
log(A(x)/x) = exp(x)*x/2 + d/dx exp(2*x)*x^3/2!/4 + d^2/dx^2 exp(3*x)*x^5/3!/8 + d^3/dx^3 exp(4*x)*x^7/4!/16 +...
Ordinary Generating Function:
O.g.f.: x + x^2 + 6*x^3 + 51*x^4 + 610*x^5 + 9360*x^6 + 175371*x^7 +...
O.g.f.: x + 2*x^2/(1-x)^3/2 + 6*2!*x^3/(1-2*x)^5/4 + 20*3!*x^4/(1-3*x)^7/8 + 70*4!*x^5/(1-4*x)^9/16 + 252*5!*x^6/(1-5*x)^11/32 +...+ (2*n)!/n!*x^(n+1)/(1-n*x)^(2*n+1)/2^n +...

Cf. A213643.

Rest[CoefficientList[InverseSeries[Series[x - x^2*E^x/2, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Dec 28 2013 *)

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

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

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, exp(m*x+x*O(x^n))/2^m*x^(2*m)/m!)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, exp(m*x+x*O(x^n))/2^m*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

/* O.g.f.: */
{a(n)=polcoeff(sum(m=0, n, (2*m)!/m!/2^m*x^(m+1)/(1-m*x+x*O(x^n))^(2*m+1)), n)}

E.g.f.: Series_Reversion(x - x^2*exp(x)/2).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)/2^n*x^(2*n) / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)/2^n*x^(2*n-1) / n! ).
O.g.f.: Sum_{n>=0} (2*n)!/n!/2^n * x^(n+1) / (1 - n*x)^(2*n+1).
a(n) = Sum_{k=0..n-1} k^(n-k-1)/(n-k-1)! * (n+k-1)!/k!/2^k.
a(n) ~ r*sqrt((1+r)/(2+r*(4+r))) * n^(n-1) / (exp(n) * (r*(1+r)/(2+r))^n), where r = 0.4912251835444738797155054345... is the root of the equation r*(2+r)*exp(r) = 2. - Vaclav Kotesovec, Dec 28 2013

A380841 Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 21, 10, 3, 1, 0, 148, 66, 18, 4, 1, 0, 1305, 560, 141, 28, 5, 1, 0, 13806, 5770, 1380, 252, 40, 6, 1, 0, 170401, 69852, 16095, 2776, 405, 54, 7, 1, 0, 2403640, 970886, 217458, 35940, 4940, 606, 70, 8, 1, 0, 38143377, 15228880, 3335745, 533304, 70045, 8088, 861, 88, 9, 1

Offset: 0

Stefano Spezia, Feb 05 2025

			Array begins as:
  1,    1,    1,     1,     1,     1,      1, ...
  0,    1,    2,     3,     4,     5,      6, ...
  0,    4,   10,    18,    28,    40,     54, ...
  0,   21,   66,   141,   252,   405,    606, ...
  0,  148,  560,  1380,  2776,  4940,   8088, ...
  0, 1305, 5770, 16095, 35940, 70045, 124350, ...
  ...

Cf. A380843 (antidiagonal sums).
Columns k=0..4 give A000007, A006153, A377529, A377530, A379993.
Rows n=0..2 give A000012, A001477, A028552.
Main diagonal gives A380842.
A(n,n+1) gives A213643(n+1).

A[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

A(n,k) = n! * Sum_{j=0..n} j^(n-j) * binomial(j+k-1,j)/(n-j)!. - Seiichi Manyama, Feb 06 2025

A215363 E.g.f. satisfies: A(x) = x + A(x)^2*cosh(A(x)).

Original entry on oeis.org

1, 2, 12, 132, 2040, 40350, 974400, 27805736, 915505920, 34160797530, 1424581678080, 65660547312492, 3314551571595264, 181866769617012662, 10777121944589844480, 685937077729538151120, 46668919680893409361920, 3380042082757952844150066, 259638732115410022642483200

Offset: 1

Paul D. Hanna, Aug 08 2012

nonn

			E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
Series expressions:
A(x) = x + cosh(x)*x^2 + d/dx cosh(x)^2*x^4/2! + d^2/dx^2 cosh(x)^3*x^6/3! + d^3/dx^3 cosh(x)^4*x^8/4! +...
log(A(x)/x) = cosh(x)*x + d/dx cosh(x)^2*x^3/2! + d^2/dx^2 cosh(x)^3*x^5/3! + d^3/dx^3 cosh(x)^4*x^7/4! +...

Cf. A215364, A213643.

Rest[CoefficientList[InverseSeries[Series[x - x^2*Cosh[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

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

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^m*x^(2*m)/m!)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^m*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

{a(n)=n!*polcoeff(x/(1 - x*cosh(x+x*O(x^n)))^n/n, n)}

E.g.f. satisfies:
(1) Series_Reversion(x - x^2*cosh(x)).
(2) x + Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^n*x^(2*n) / n!.
(3) x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^n*x^(2*n-1) / n! ).
a(n) = [x^n/n!] x/(1 - x*cosh(x))^n / n for n>0.
a(n) ~ s*sqrt(r/(6*r-2*s-r*s^2+s^3)) * n^(n-1) / (exp(n) * r^n), where s = 0.4227473416936597149... is the root of the equation s*(2*cosh(s) + s*sinh(s)) = 1, and r = s - s^2*cosh(s) = 0.22782318947143997934... - Vaclav Kotesovec, Jan 13 2014

A380842 Main diagonal of the array A380841.

Original entry on oeis.org

1, 1, 10, 141, 2776, 70045, 2157156, 78452521, 3290644288, 156380715801, 8304267312100, 487328231729581, 31318669850761008, 2187567259278425557, 165011952533314548676, 13368463736048341225425, 1157693100510102752463616, 106719312722496774534400177, 10433609651067618426072766020

Offset: 0

Stefano Spezia, Feb 05 2025

nonn

Cf. A213643, A380841.

A380841[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A380841[n,n],{n,0,18}]

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+n-1, k)/(n-k)!); \\ Seiichi Manyama, Feb 06 2025

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+n-1,k)/(n-k)!. - Seiichi Manyama, Feb 06 2025

a(n) ~ r^(n + 1/2) * (1+r)^n * n^n / (sqrt(1 + 2*r - r^2) * exp(n) * (1-r)^n), where r = 0.760359234033398901446642379997259705906638343193092252797... is the root of the equation exp(1-r)*(1-r^2)^r = r^(2*r). - Vaclav Kotesovec, May 29 2025

A214363 E.g.f. satisfies: A(x) = x + A(x)^2 * cosh(A(x))^2 / 2.

Original entry on oeis.org

1, 1, 3, 27, 285, 3585, 56595, 1062131, 22868685, 557624745, 15204727395, 458112683787, 15113457195837, 541914801559313, 20984168325697395, 872681528769576675, 38793582477781496685, 1835683831177469267769, 92124361183712633639235, 4887330703061330205124475

Offset: 1

Paul D. Hanna, Jul 13 2012

nonn

a(n) (mod 3) yields period 6 sequence: [0,0,0,0,0,2] starting at n=3.

a(6*n+2) == 2 (mod 3) for n>=1.

			E.g.f: A(x) = x + x^2/2! + 3*x^3/3! + 27*x^4/4! + 285*x^5/5! + 3585*x^6/6! +...
Related expansions:
A(x)^2 = 2*x^2/2! + 6*x^3/3! + 30*x^4/4! + 330*x^5/5! + 4410*x^6/6! + 67830*x^7/7! +...
A(x) = x + cosh(x)^2*x^2/2 + d/dx cosh(x)^4*x^4/(4*2!) + d^2/dx^2 cosh(x)^6*x^6/(8*3!) + d^3/dx^3 cosh(x)^8*x^8/(16*4!) +...
log(A(x)/x) = 1 + cosh(x)^2*x/2 + d/dx cosh(x)^4*x^3/(4*2!) + d^2/dx^2 cosh(x)^6*x^5/(8*3!) + d^3/dx^3 cosh(x)^8*x^7/(16*4!) +...

Cf. A213643.

Rest[CoefficientList[InverseSeries[Series[x - (x^2*Cosh[x]^2)/2,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 12 2014 *)

{a(n)=n!*polcoeff(serreverse(x-x^2/2*cosh(x+x*O(x^n))^2),n)}
for(n=1,30,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^(2*m)*x^(2*m)/2^m/m!)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^(2*m)*x^(2*m-1)/2^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

E.g.f. satisfies:
(1) A(x - x^2*cosh(x)^2/2) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^(2*n)*x^(2*n) / (2^n*n!).
(2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^(2*n)*x^(2*n-1) / (2^n*n!) ).
a(n) ~ n^(n-1) * sqrt(2/(1 + (1+2*s^2)*cosh(2*s) + 4*s*sinh(2*s))) / (exp(n) * r^(n-1/2)), where s = 0.568824148293580379787367453... is the root of the equation s*cosh(s)*(cosh(s) + s*sinh(s)) = 1, and r = s - s^2*(cosh(s))^2/2 = 0.3488028982491643456675... - Vaclav Kotesovec, Jan 12 2014

A227462 E.g.f. equals the series reversion of x - x^2*exp(x^2).

Original entry on oeis.org

1, 2, 12, 144, 2400, 50760, 1310400, 39984000, 1407490560, 56143130400, 2502730137600, 123302540148480, 6653043673436160, 390182090889951360, 24713127865269043200, 1681180295452142284800, 122252619549518954496000, 9463442426016057083404800, 776944497581062575154790400

Offset: 1

Paul D. Hanna, Jul 13 2013

nonn

Limit n->infinity (a(n)/n!)^(1/n) = Limit n->infinity a(n+1)/a(n)/n = 4.6910715... = 2*(1+s^2)/(s*(1+2*s^2)), where s is defined below. - Vaclav Kotesovec, added Jul 28 2013, updated Jan 07 2014

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 144*x^4/4! + 2400*x^5/5! +...
where A(x) = x + A(x)^2*exp(A(x)^2).
The e.g.f. satisfies:
(3) A(x) = x + x^2*exp(A(x)^2) + 2*x^3*exp(2*A(x)^2) + 5*x^4*exp(3*A(x)^2) + 14*x^5*exp(4*A(x)^2) + 42*x^6*exp(5*A(x)^2) +...
(4) log(A(x)/x) = x*exp(A(x)^2) + 3*x^2*exp(2*A(x)^2)/2 + 10*x^3*exp(3*A(x)^2)/3 + 35*x^4*exp(4*A(x)^2)/4 + 126*x^5*exp(5*A(x)^2)/5 +...
(5) A(x) = x + x^2/2*exp(x^2) + d/dx x^4/4*exp(2*x^2)/2! + d^2/dx^2 x^6/8*exp(3*x^2)/3! + d^3/dx^3 x^8/16*exp(4*x^2)/4! +...
(6) log(A(x)/x) = x*exp(x^2)/2 + d/dx x*exp(2*x^2)/2! + d^2/dx^2 x^2*exp(3*x^2)/3! + d^3/dx^3 x^3*exp(4*x^2)/4! +...

Cf. A213643, A227463, A000108.

Table[(n-1)!*SeriesCoefficient[(x/(x-x^2*E^(x^2)))^n,{x,0,n-1}],{n,1,20}] (* Vaclav Kotesovec, Jul 28 2013 *)

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

/* E.g.f. A(x) = x*Catalan( x*exp(A(x)^2) ): */
{a(n)=local(A=x); for(i=1,n,A=(1-sqrt(1-4*x*exp(A^2 +x^2*O(x^n)) ))/2*exp(-A^2 +x*O(x^n)) ); n!*polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*exp(x^2+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*exp(x^2+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))

E.g.f. A(x) satisfies:
(1) A(x) = x + A(x)^2*exp(A(x)^2).
(2) A(x) = x*Catalan( x*exp(A(x)^2) ) where Catalan(x) = (1-sqrt(1-4*x))/(2*x).
(3) A(x) = x*Sum_{n>=0} binomial(2*n+1,n)/(2*n+1) * x^n * exp(n*A(x)^2).
(4) A(x) = x*exp( Sum_{n>=1} binomial(2*n-1,n) * x^n/n * exp(n*A(x)^2) ).
(5) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * exp(n*x^2) / n!.
(6) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * exp(n*x^2) / n! ).
a(n) ~ n^(n-1) * s * sqrt((1+2*s^2)/(2+10*s^2+4*s^4)) / (exp(n) * ((s*(1+2*s^2))/(2*(1+s^2)))^n), where s = 0.3788063540000847107637564... is the root of the equation 2*s*(1+s^2)*exp(s^2) = 1. - Vaclav Kotesovec, Jan 07 2014

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A161629 E.g.f. satisfies: A(x) = exp( x*Catalan(x*A(x)) ), where Catalan(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A143136 E.g.f. satisfies: A(x) = x + sinh( A(x) )^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

A214688 E.g.f. equals the series reversion of x - x^2*exp(2*x).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

PARI

PARI

SageMath

Formula

Extensions

A215003 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x))/2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

PARI

Formula

A380841 Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

A161629 E.g.f. satisfies: A(x) = exp( xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2x) is the g.f. of A000108.

A214688 E.g.f. equals the series reversion of x - x^2exp(2x).