A213644 -id:A213644 - cp's OEIS frontend

A161633 E.g.f. satisfies A(x) = 1/(1 - xexp(xA(x))).

1, 1, 4, 27, 268, 3525, 57966, 1146061, 26500552, 702069129, 20974309210, 697754762001, 25584428686620, 1025230366195789, 44579963354153878, 2090676600895922565, 105191995364927688976, 5652501986238910061073, 323083811850594613809714, 19573120681427758058921881

Offset: 0

Paul D. Hanna, Jun 18 2009

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3525*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...

Seiichi Manyama, Table of n, a(n) for n = 0..374

Cf. A006153, A161630 (e.g.f. = exp(x*A(x))), A213644, A364980, A364981.

Cf. A366232, A366233, A366234, A366235.

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

a(n,m=1)=n!*sum(k=0,n,binomial(n+m,k)*m/(n+m)*k^(n-k)/(n-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(x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).
(3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.
a(n) = n! * Sum_{k=0..n} binomial(n+1,k)/(n+1) * k^(n-k)/(n-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,k)*m/(n+m) * k^(n-k)/(n-k)!.
a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - Vaclav Kotesovec, Jan 10 2014

A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4xexp(x))).

Original entry on oeis.org

1, 1, 6, 57, 796, 14785, 344046, 9640225, 316255416, 11896233345, 504918768250, 23874754106401, 1244712973780068, 70940791877082049, 4388291507415513894, 292823509879910802465, 20966854494419642792176, 1603540841320336494905089, 130464295561360336835272050

Offset: 0

Ilya Gutkovskiy, Nov 18 2017

nonn

Inverse binomial transform of A194471.

Seiichi Manyama, Table of n, a(n) for n = 0..356

Cf. A000108, A006531, A052895, A194471, A213644, A295239.

a:=series(2/(1+sqrt(1-4*x*exp(x))),x=0,19): seq(n!*coeff(a,x,n),n=0..18); # Paolo P. Lava, Mar 27 2019

nmax = 18; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 18}]

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Aug 15 2023

E.g.f.: 1/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2*(1 + LambertW(1/4))) * n^(n-1) / ((LambertW(1/4))^n * exp(n)). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+1,k)/( (2*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Aug 15 2023

A213643 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x)).

Original entry on oeis.org

1, 2, 18, 252, 4940, 124350, 3823722, 138915560, 5822192952, 276522143130, 14677209803630, 860990013672492, 55315008281020644, 3862656545279925302, 291301089508829138130, 23595204076694940812880, 2042970533426395737658352, 188298566037963463789282482

Offset: 1

Paul D. Hanna, Jun 17 2012

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 252*x^4/4! + 4940*x^5/5! +...
where A(x - x^2*exp(x)) = x and A(x) = x + A(x)^2*exp(A(x)).
Related expansions:
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 168*x^4/4! + 3240*x^5/5! + 80880*x^6/6! +...
A(x) = x*Catalan(x*G(x)) where G(x) = exp(A(x)):
exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! + 171421*x^6/6! +..., which is the e.g.f. of A161629.
A(x) = x + exp(x)*x^2 + d/dx exp(2*x)*x^4/2! + d^2/dx^2 exp(3*x)*x^6/3! + d^3/dx^3 exp(4*x)*x^8/4! +...
log(A(x)/x) = exp(x)*x + d/dx exp(2*x)*x^3/2! + d^2/dx^2 exp(3*x)*x^5/3! + d^3/dx^3 exp(4*x)*x^7/4! +...
Ordinary Generating Function:
O.g.f.: x + 2*x^2 + 18*x^3 + 252*x^4 + 4940*x^5 + 124350*x^6 +...
O.g.f.: x + 2*x^2/(1-x)^3 + 6*2!*x^3/(1-2*x)^5 + 20*3!*x^4/(1-3*x)^7 + 70*4!*x^5/(1-4*x)^9 + 252*5!*x^6/(1-5*x)^11 +...+ (2*n)!/n!*x^(n+1)/(1-n*x)^(2*n+1) +...

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

Cf. A213644, A161629, A200319, A215003.

a:= n-> n!*coeff(series(RootOf(A=x+A^2*exp(A), A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 18 2013

Flatten[{1,Table[Sum[k^(n-k-1)/(n-k-1)!*(n+k-1)!/k!,{k,0,n-1}],{n,2,20}]}] (* Vaclav Kotesovec, Jul 13 2013 *)

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

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

E.g.f.: A(x) = log(G(x)) where G(x) = exp(x*Catalan(x*G(x))) is the e.g.f. of A161629, and Catalan(x) = (1-sqrt(1-4*x))/(2*x).
E.g.f.: Series_Reversion(x - x^2*exp(x)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)*x^(2*n) / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)*x^(2*n-1) / n! ).
O.g.f.: Sum_{n>=0} (2*n)!/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!.
a(n) = n*A213644(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). - Vaclav Kotesovec, Jul 13 2013
a(n) ~ (1-r) * n^(n-1) * (r*(1+r)/(1-r))^n / (sqrt(r*(1+2*r-r^2))*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

A377546 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(x))^2 ).

Original entry on oeis.org

1, 2, 18, 294, 7136, 231410, 9421932, 462459242, 26593896912, 1754278123266, 130611457831700, 10835721949072922, 991315043401627320, 99154012317212577218, 10765112531819005907484, 1260860266373297376720810, 158473050112495481401395872, 21275613503385328981848681986

Offset: 0

Seiichi Manyama, Oct 31 2024

nonn

Index entries for reversions of series

Cf. A213644, A377548.

Cf. A364985.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*exp(x))^2)/x))

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

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364985.
a(n) = 2 * n! * Sum_{k=0..n} k^(n-k) * binomial(2*n+k+2,k)/( (2*n+k+2)*(n-k)! ).

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)!.

A370928 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(x^3)) ).

Original entry on oeis.org

1, 1, 4, 30, 360, 5880, 120960, 2996280, 86889600, 2889976320, 108501724800, 4539844108800, 209497816281600, 10570762445443200, 578997352591257600, 34214810278128480000, 2169772724008976486400, 146984464202544531763200

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A213644, A370927.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*exp(x^3)))/x))

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

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

A370985 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - x^3exp(x)) ).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 3000, 45570, 403536, 10644984, 297562320, 5517833310, 142801022760, 5076208052916, 150282366476424, 4713707747551530, 189345734667052320, 7517503455423740400, 295622259241028433696, 13370535071068474177974, 642403497550155241197240

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A213644, A370984.

Cf. A358081, A370928.

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

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

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

A379688 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - 2xexp(x)) ).

Original entry on oeis.org

1, 2, 20, 366, 9992, 365130, 16769292, 929022206, 60323670416, 4494465562770, 378025706776340, 35434198578761862, 3663111561838580568, 414057463231218044186, 50805545997014472821276, 6725525908390393438264590, 955435863749903677193184032, 144987884255349864723586105122

Offset: 0

Seiichi Manyama, Dec 29 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..326

Cf. A213644, A380097.
Cf. A379659, A379687.
Cf. A380095.

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

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^(n-k) * (n-k)^k * (2*n-k)!/(k! * (n-k)!).
E.g.f. A(x) satisfies A(x) = 1/( 1 - 2*x*A(x)*exp(x*A(x)) ).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380095.

A215364 E.g.f. A(x) satisfies A(x) = 1 + xA(x)^2cosh(x*A(x)).

Original entry on oeis.org

1, 1, 4, 33, 408, 6725, 139200, 3475717, 101722880, 3416079753, 129507425280, 5471712276041, 254965505507328, 12990483544072333, 718474796305989632, 42871067358096134445, 2745230569464318197760, 187780115708775158008337, 13665196427126843296972800

Offset: 0

Paul D. Hanna, Aug 08 2012

nonn

			E.g.f: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! + 6725*x^5/5! +...

Cf. A162654, A185951, A215363, A213644.

CoefficientList[1/x*InverseSeries[Series[x-x^2*Cosh[x], {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 13 2014 *)

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

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

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+k+1,k)/(n+k+1)*a185951(n, k)); \\ Seiichi Manyama, Feb 16 2025

E.g.f. satisfies:
(1) A(x) = (1/x)*Series_Reversion(x-x^2*cosh(x)).
(2) A(x) = 1/(1 - x*A(x)*cosh(x*A(x))).
(3) A(x-x^2*cosh(x)) = 1/(1-x*cosh(x)).
a(n) = [x^n/n!] 1/(1 - x*cosh(x))^(n+1) / (n+1).
a(n) ~ n^(n-1) * s*sqrt(1/(6-2*s-r^2*s^2+r^2*s^3)) / (exp(n) * r^n), where r = 0.2278231894714399793... and s = 1.855593992316816009... are the roots of the equations r*s*(2*cosh(r*s) + r*s*sinh(r*s)) = 1, 1 + r*s^2*cosh(r*s) = s. - Vaclav Kotesovec, Jan 13 2014
a(n) = Sum_{k=0..n} k! * binomial(n+k+1,k)/(n+k+1) * A185951(n,k). - Seiichi Manyama, Feb 16 2025

cp's OEIS Frontend

A161633 E.g.f. satisfies A(x) = 1/(1 - x*exp(x*A(x))).

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A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*exp(x))).

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A213643 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x)).

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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.

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A377546 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^2 ).

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

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A370928 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x^3)) ).

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A161633 E.g.f. satisfies A(x) = 1/(1 - xexp(xA(x))).

A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4xexp(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.

A377546 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(x))^2 ).

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

A370928 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(x^3)) ).

A370985 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - x^3exp(x)) ).

A379688 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - 2xexp(x)) ).

A215364 E.g.f. A(x) satisfies A(x) = 1 + xA(x)^2cosh(x*A(x)).