cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A304866 E.g.f. A(x) satisfies: Sum_{n>=0} (n*x - A(x))^n / n! = 1.

Original entry on oeis.org

1, 1, 5, 40, 434, 5921, 97152, 1861224, 40757712, 1003975497, 27471881920, 826643389408, 27127000901376, 964082532067097, 36887864803320832, 1511803871474266800, 66072899130124806144, 3067434610357079350833, 150748671875250474491904, 7818483194884075165619704, 426760505829170289041408000, 24454432374649526694779591985, 1467784259166552629365984329728
Offset: 1

Views

Author

Paul D. Hanna, Jun 16 2018

Keywords

Examples

			E.g.f.: A(x) = x + x^2/2! + 5*x^3/3! + 40*x^4/4! + 434*x^5/5! + 5921*x^6/6! + 97152*x^7/7! + 1861224*x^8/8! + 40757712*x^9/9! + 1003975497*x^10/10! + ...
such that
1  =  1 + (x - A(x)) + (2*x - A(x))^2/2! + (3*x - A(x))^3/3! + (4*x - A(x))^4/4! + (5*x - A(x))^5/5! + (6*x - A(x))^6/6! + (7*x - A(x))^7/7! + ...
Also,
-LambertW(-x)  =  A(x) + A(x)*(2*x - A(x))/2! + A(x)*(3*x - A(x))^2/3! + A(x)*(4*x - A(x))^3/4! + A(x)*(5*x - A(x))^4/5! + A(x)*(6*x - A(x))^5/6! + ...  =  x + 2*x^2/2! + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + 6^5*x^6/6! + ...
Further,
LambertW(-x)/(-x)  =  1 + (2*x - A(x)) + (3*x - A(x))^2/2! + (4*x - A(x))^3/3! + (5*x - A(x))^4/4! + (6*x - A(x))^5/5! + (7*x - A(x))^6/6! + ...  =  1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + 7^5*x^6/6! + ...
Note the following series relations involving e.g.f. A(x):
Sum_{n>=0} -A(x) * (n*x - A(x))^(n-1) / n!  =  1 - Sum_{n>=1} n^(n-1) * x^n / n!  =  1 + LambertW(-x),
Sum_{n>=0} ((n + m)*x - A(x))^n / n!  =  Sum_{n>=0} m*(n + m)^(n-1) * x^n / n!  =  ( LambertW(-x)/(-x) )^m.
These relations imply
( LambertW(-x) / (-x) )^(-A(x)/x)  =  1 + LambertW(-x)
giving A(x) = log(1 + LambertW(-x)) * x / LambertW(-x).

Links

Crossrefs

Cf. A300014 (exp(A(x))), A306090.

Programs

  • Mathematica
    nmax = 20; Rest[CoefficientList[Series[Log[1 + LambertW[-x]] * x / LambertW[-x], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Sep 01 2020 *)
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, (m*x - x*Ser(A))^m/m!), #A)); n!*A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
(1) Sum_{n>=0} (n*x - A(x))^n / n! = 1.
(2) Sum_{n>=0} (n*x - 2*A(x))^n / n! = 1 + LambertW(-x).
(3) Sum_{n>=0} (n*x - m*A(x))^n / n! = (1 + LambertW(-x))^(m-1).
(4) Sum_{n>=0} -A(x) * (n*x - A(x))^(n-1) / n! = 1 + LambertW(-x).
(5) Sum_{n>=0} ((n + 1)*x - A(x))^n / n! = LambertW(-x)/(-x).
(6) Sum_{n>=0} ((n + m)*x - A(x))^n / n! = ( LambertW(-x)/(-x) )^m.
(7) Sum_{n>=0} ((n + m)*x + p*A(x))^n / n! = ( LambertW(-x)/(-x) )^m / (1 + LambertW(-x))^(p+1).
(8) ( LambertW(-x) / (-x) )^(-A(x)/x) = 1 + LambertW(-x).
(9) A(x) = log(1 + LambertW(-x)) * x / LambertW(-x).
a(n) ~ sqrt(Pi/2) * exp(-1) * n^(n - 1/2) * (1 - log(n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, Sep 01 2020

A306066 E.g.f. A(x) satisfies: Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k)*x + (k+1)*A(x) = 0.

Original entry on oeis.org

-1, 2, -6, 36, -300, 3270, -43680, 691992, -12670560, 263281050, -6119720640, 157325242140, -4431909081600, 135757694361198, -4492575720622080, 159723265791222000, -6071451523596103680, 245720759937001346610, -10548874580411105832960, 478801529559871868317140, -22909292930454154076160000, 1152457216135660417348971990, -60807227650606789798265487360, 3357825559218695417748978547080
Offset: 1

Views

Author

Paul D. Hanna, Jun 27 2018

Keywords

Comments

A306065(n-1) = (-1)^n * a(n)/n for n >= 1.

Examples

			E.g.f.: A(x) = -x + 2*x^2/2! - 6*x^3/3! + 36*x^4/4! - 300*x^5/5! + 3270*x^6/6! - 43680*x^7/7! + 691992*x^8/8! - 12670560*x^9/9! + 263281050*x^10/10! - 6119720640*x^11/11! + 157325242140*x^12/12! - 4431909081600*x^13/13! + 135757694361198*x^14/14! - 4492575720622080*x^15/15! + 159723265791222000*x^16/16! + ...
such that
0 = (x + A(x)) + (2*x + A(x))*(x + 2*A(x))/1! + (3*x + A(x))*(2*x + 2*A(x))*(x + 3*A(x))/2! + (4*x + A(x))*(3*x + 2*A(x))*(2*x + 3*A(x))*(x + 4*A(x))/3! + (5*x + A(x))*(4*x + 2*A(x))*(3*x + 3*A(x))*(2*x + 4*A(x))*(x + 5*A(x))/4! + ...
Also,
1 = 1 + (x + A(x))/(1!*2) + (3*x + A(x))*(x + 3*A(x))/(2!*2^2) + (5*x + A(x))*(3*x + 3*A(x))*(x + 5*A(x))/(3!*2^3) + (7*x + A(x))*(5*x + 3*A(x))*(3*x + 5*A(x))*(x + 7*A(x))/(4!*2^4) + ...
EXAMPLES OF SUMS.
The e.g.f. A(x) satisfies the following sums.
(E1) Define
S1(m,p) = Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k - m)*x + (k+1 - p)*A(x),
then
S1(m,p) = -(m*x + p*A(x)) * A(x)^(2*m-2) * x^(2*p-2) / (x + A(x))^(m+p-2).
(E2) Define
S2(m,p) = Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1 - m)*x + (2*k-1 - p)*A(x),
then
S2(m,p) = (-A(x)/x)^( (m*x + p*A(x))/(x - A(x)) ).
Examples of S2(m,p):
S2(0,0) = 1,
S2(1,-1) = -A(x)/x,
S2(-1,1) = -x/A(x),
S2(1,2) =  (-A(x)/x)^( (x + 2*A(x))/(x - A(x)) ).
Examples of S1(m,p):
S1(0,0) = 0,
S1(1,0) = -(x + A(x)) / x,
S1(2,0) = -2*A(x)^2 / x,
S1(3,0) = -3*A(x)^4 / (x*(x  + A(x))),
S1(0,1) = -(x + A(x)) / A(x),
S1(0,2) = -2*x^2 / A(x),
S1(0,3) = -3*x^4 / (A(x)*(x + A(x))),
S1(1,1) = -(x + A(x)),
S1(2,2) = -2*x^2*A(x)^2 / (x + A(x)),
S1(3,3) = -3*x^4*A(x)^4 / (x + A(x))^3,
S1(2,1) = -A(x)^2 * (2*x + A(x)) / (x + A(x)),
S1(1,2) = -x^2 * (x + 2*A(x)) / (x + A(x)),
S1(4,3) = -x^4*A(x)^6 * (4*x + 3*A(x)) / (x + A(x))^5.

Links

Crossrefs

Cf. A306090, A306067, A306065.

Programs

  • PARI
    {a(n) = my(A=[-1]); for(i=0,n, A=concat(A,0); A[#A] = -polcoeff( sum(m=0, #A, 1/m!*prod(k=0, m, (m+1-k)*x + (k+1)*x*Ser(A) ) ), #A)); n!*A[n]}
for(n=1,30, print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k)*x + (k+1)*A(x) = 0.
(2) Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k - m)*x + (k+1)*A(x) = -m/x * A(x)^(2*m-2) / (x + A(x))^(m-2).
(3) Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k)*x + (k+1 - p)*A(x) = -p/A(x) * x^(2*p-2) / (x + A(x))^(p-2).
(4) Sum_{n>=0} 1/n! * Product_{k=0..n} (n+1-k - m)*x + (k+1 - p)*A(x) = -(m*x + p*A(x)) * A(x)^(2*m-2) * x^(2*p-2) / (x + A(x))^(m+p-2).
(5) A(A(x)) = x.
(6) Sum_{n>=0} 1/n! * Product_{k=1..n} (n-k)*x + k*A(x) = -A(x)/x.
(7) Sum_{n>=0} 1/n! * Product_{k=1..n} (n-k+1)*x + (k-1)*A(x) = -x/A(x).
(8) Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1)*x + (2*k-1)*A(x) = 1.
(9) Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1 - m)*x + (2*k-1)*A(x) = (-A(x)/x)^( m*x/(x - A(x)) ).
(10) Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1)*x + (2*k-1 - p)*A(x) = (-A(x)/x)^( p*A(x)/(x - A(x)) ).
(11) Sum_{n>=0} 1/(n!*2^n) * Product_{k=1..n} (2*(n-k)+1 - m)*x + (2*k-1 - p)*A(x) = (-A(x)/x)^( (m*x + p*A(x))/(x - A(x)) ).
a(n)/n! ~ (-1)^n * c * d^n / n^(3/2), where d = 2.45598128882155545489... and c = 0.2658048623687886... - Vaclav Kotesovec, Jul 12 2018

A306067 E.g.f. A(x) satisfies: Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + (k+1)*A(x) = 1.

Original entry on oeis.org

1, 4, 21, 178, 2279, 39066, 835132, 21400198, 640239525, 21920851282, 845615003996, 36298192983482, 1716348366690487, 88653661788525666, 4967006270867149524, 300043327305644202366, 19440451816128996788777, 1344909407655243937857826, 98949254253416815493778796, 7714902418308597200477578514, 635444724815621395463510504211, 55134789286331454820101232131938
Offset: 0

Views

Author

Paul D. Hanna, Jun 25 2018

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 21*x^2/2! + 178*x^3/3! + 2279*x^4/4! + 39066*x^5/5! + 835132*x^6/6! + 21400198*x^7/7! + 640239525*x^8/8! + 21920851282*x^9/9! + 845615003996*x^10/10! + ...
such that
1  =  (0 + A(x))  -  (1 + A(x))*(0 + 2*A(x))*x  +  (2 + A(x))*(1 + 2*A(x))*(0 + 3*A(x))*x^2/2!  -  (3 + A(x))*(2 + 2*A(x))*(1 + 3*A(x))*(0 + 4*A(x))*x^3/3!  +  (4 + A(x))*(3 + 2*A(x))*(2 + 3*A(x))*(1 + 4*A(x))*(0 + 5*A(x))*x^4/4!  -  (5 + A(x))*(4 + 2*A(x))*(3 + 3*A(x))*(2 + 4*A(x))*(1 + 5*A(x))*(0 + 6*A(x))*x^5/5!  +  ...

Links

Crossrefs

Cf. A306090.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = -Vec( sum(n=0,#A, (-x)^n/n!* prod(k=0,n, (n-k) + (k+1)*Ser(A) ) ) )[#A] ); n!*A[n+1]}
for(n=0,20, print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + k*A(x) = -x.
(2) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k) + (k+1)*A(x) = 1.
(3) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n-k+1) + k*A(x) = 1/A(x).
a(n)/n! ~ c * d^n / n^(3/2), where d = 4.423034555284689... and c = 3.17922741818... - Vaclav Kotesovec, Jul 12 2018

A306089 G.f. A(x) satisfies: Sum_{n>=0} (-1)^n * Product_{k=1..n} x^(n+1-k) + (-A(x))^k = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 21, 68, 186, 495, 1335, 3744, 10870, 32120, 95565, 284830, 850580, 2548436, 7669604, 23192434, 70443076, 214768128, 656857897, 2014416494, 6192794179, 19081689920, 58923909932, 182331403224, 565289067360, 1755737915942, 5462257817753, 17019938788706, 53109742992332, 165952503650622, 519222849063545, 1626498619326355, 5100995860701418
Offset: 1

Views

Author

Paul D. Hanna, Jun 21 2018

Keywords

Examples

			G.f.: A(x) = x + x^2 + x^3 + x^4 + x^5 + 6*x^6 + 21*x^7 + 68*x^8 + 186*x^9 + 495*x^10 + 1335*x^11 + 3744*x^12 + 10870*x^13 + 32120*x^14 + 95565*x^15 + ...
such that
1  =  1  -  (x - A(x))  +  (x + A(x)^2)*(x^2 - A(x))  -  (x - A(x)^3)*(x^2 + A(x)^2)*(x^3 - A(x))  +  (x + A(x)^4)*(x^2 - A(x)^3)*(x^3 + A(x)^2)*(x^4 - A(x))  -  (x - A(x)^5)*(x^2 + A(x)^4)*(x^3 - A(x)^3)*(x^4 + A(x)^2)*(x^5 - A(x)) + ...

Links

Crossrefs

Cf. A306088, A306090.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = -Vec( sum(m=0,#A, (-1)^m * prod(k=1,m, x^(m+1-k) + (-x)^k*Ser(A)^k ) ) )[#A+1]); A[n]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) satisfies: A(-A(-x)) = x.

A306091 G.f. A(x) satisfies: (1 + A(x))^A(x) = (1 + x)^x ; this sequence gives the denominators of the coefficients of x^n in g.f. A(x).

Original entry on oeis.org

1, 2, 4, 6, 8, 1440, 960, 120960, 48384, 7257600, 1612800, 479001600, 4561920, 5230697472000, 10461394944000, 7846046208000, 6974263296000, 9146248151040000, 8536498274304000, 1502674769756160000, 1857852442607616000, 67440043666656460800000, 44960029111104307200000, 18613452051997183180800000, 954536002666522214400000
Offset: 1

Views

Author

Paul D. Hanna, Jun 21 2018

Keywords

Comments

The numerators of the coefficients in g.f. A(x) are given by A306090.

Examples

			G.f.: A(x) = -x + 1/2*x^2 - 1/4*x^3 + 1/6*x^4 - 1/8*x^5 + 143/1440*x^6 - 79/960*x^7 + 8483/120960*x^8 - 2953/48384*x^9 + 391753/7257600*x^10 - 77983/1612800*x^11 + 20963473/479001600*x^12 - 182269/4561920*x^13 + 192178874539/5230697472000*x^14 - 355629691849/10461394944000*x^15 + 248105704337/7846046208000*x^16 - 206101262483/6974263296000*x^17 + 253628381647657/9146248151040000*x^18 - 222936799599583/8536498274304000*x^19 + 37078279922025269/1502674769756160000*x^20 + ... + A306090(n)/A306091(n)*x^n + ...
such that
(E.1) 1  =  1  +  (x + A(x))  +  (x + 2*A(x))*(2*x + A(x))/2!  +  (x + 3*A(x))*(2*x + 2*A(x))*(3*x + A(x))/3!  +  (x + 4*A(x))*(2*x + 3*A(x))*(3*x + 2*A(x))*(4*x + A(x))/4!  +  (x + 5*A(x))*(2*x + 4*A(x))*(3*x + 3*A(x))*(4*x + 2*A(x))*(5*x + A(x))/5! + ...
(E.2) (1 + x)^p  =  1  +  (x + (1-p)*A(x))  +  (x + (2-p)*A(x))*(2*x + (1-p)*A(x))/2!  +  (x + (3-p)*A(x))*(2*x + (2-p)*A(x))*(3*x + (1-p)*A(x))/3!  +  (x + (4-p)*A(x))*(2*x + (3-p)*A(x))*(3*x + (2-p)*A(x))*(4*x + (1-p)*A(x))/4! + ...
(E.3) (1 + A(x))^m  =  1  +  ((1-m)*x + A(x))  +  ((1-m)*x + 2*A(x))*((2-m)*x + A(x))/2!  +  ((1-m)*x + 3*A(x))*((2-m)*x + 2*A(x))*((3-m)*x + A(x))/3!  +  ((1-m)*x + 4*A(x))*((2-m)*x + 3*A(x))*((3-m)*x + 2*A(x))*((4-m)*x + A(x))/4! + ...
FUNCTIONAL EQUATIONS.
The series A(x) satisfies:
(E.4) (1 + A(x))^A(x) = (1 + x)^x  =  1 + x^2 - 1/2*x^3 + 5/6*x^4 - 3/4*x^5 + 33/40*x^6 - 5/6*x^7 + 2159/2520*x^8 - 209/240*x^9 + ...
GENERATING METHOD.
Although the functional equation (1 + A(x))^A(x) = (1 + x)^x has an infinite number of solutions, one may arrive at the g.f. A(x) by the following iteration.
If we start with A = -x, and iterate
(E.5) A = (A + x*log(1 + x)/log(1 + A))/2
then A will converge to g.f. A(x).

Links

Crossrefs

Cf. A306090 (numerators).

Programs

  • Mathematica
    nmax = 25; sol = {a[1] -> -1};
Do[A[x_] = Sum[a[k] x^k, {k, 1, n}] /. sol; eq = CoefficientList[(1 + A[x])^A[x] - (1 + x)^x + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax + 1}];
sol /. Rule -> Set;
a /@ Range[1, nmax] // Denominator (* Jean-François Alcover, Nov 02 2019 *)
  • PARI
    /* From Functional Equation (1 + A(x))^A(x) = (1 + x)^x */
{a(n) = my(A = -x +x*O(x^n)); for(i=1,n, A = (A + x*log(1+x +x*O(x^n))/log(1+A))/2 ); denominator( polcoeff(A,n) )}

Formula

G.f. A(x) = Sum_{n>=0} A306090(n)/A306091(n) * x^n satisfies:
(1) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + k*A(x) = 1.
(2) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + (k - p)*A(x) = (1 + x)^p.
(3) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + k*A(x) = (1 + A(x))^m.
(4) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + (k - p)*A(x) = (1+x)^p * (1 + A(x))^m.
(5) A(A(x)) = x.
(6) (1 + A(x))^A(x) = (1 + x)^x.
(7) Sum_{n>=1} (-A(x))^(n+1) / n = x*log(1+x).
(8) Let F(x,y) = Series_Reversion( (exp(-x*y) - exp(-x))/(1-y) ), where the inverse is taken wrt x, and let F'(x,y) = d/dx F(x,y), then F'(x, A(x)/x) = 1 (derived from Peter Bala's g.f. for A067948).

A306065 E.g.f. A(x) satisfies: Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k) - (k+1)*A(x) = 0.

Original entry on oeis.org

1, 1, 2, 9, 60, 545, 6240, 86499, 1407840, 26328105, 556338240, 13110436845, 340916083200, 9696978168657, 299505048041472, 9982704111951375, 357144207270359040, 13651153329833408145, 555203925284795043840, 23940076477993593415857, 1090918710974007336960000, 52384418915257291697680545
Offset: 0

Views

Author

Paul D. Hanna, Jun 30 2018

Keywords

Comments

a(n) = (-1)^(n+1) * A306066(n+1)/(n+1) for n >= 0.

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 60*x^4/4! + 545*x^5/5! + 6240*x^6/6! + 86499*x^7/7! + 1407840*x^8/8! + 26328105*x^9/9! + 556338240*x^10/10! + ...
such that
0  =  (1 - A(x)) - (2 - A(x))*(1 - 2*A(x))*x/1! + (3 - A(x))*(2 - 2*A(x))*(1 - 3*A(x))*x^2/2! - (4 - A(x))*(3 - 2*A(x))*(2 - 3*A(x))*(1 - 4*A(x))*x^3/3! + (5 - A(x))*(4 - 2*A(x))*(3 - 3*A(x))*(2 - 4*A(x))*(1 - 5*A(x))*x^4/4! + ...
Also,
1  =  1 - (1 - A(x))*x/(1!*2) + (3 - A(x))*(1 - 3*A(x))*x^2/(2!*2^2) - (5 - A(x))*(3 - 3*A(x))*(1 - 5*A(x))*x^3/(3!*2^3) + (7 - A(x))*(5 - 3*A(x))*(3 - 5*A(x))*(1 - 7*A(x))*x^4/(4!*2^4) - (9 - A(x))*(7 - 3*A(x))*(5 - 5*A(x))*(3 - 7*A(x))*(1 - 9*A(x))*x^5/(5!*2^5) + ...
More generally, the e.g.f. A(x) satisfies the following sums.
Define
S1(m,p) = Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k - m) - (k+1 - p)*A(x),
then
S1(m,p) = -(m - p*A(x)) * A(x)^(2*m-2) * x^(m+p-2) / (A(x) - 1)^(m+p-2).
Define
S2(m,p) = Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1 - m) - (2*k-1 - p)*A(x),
then
S2(m,p) = A(x)^( (m - p*A(x))/(1 + A(x)) ).
RELATED SERIES.
The e.g.f. also satisfies A(x) = 1/A(-x*A(x)), where:
A(-x*A(x)) = 1/A(x) = 1 - x - 3*x^3/3! - 12*x^4/4! - 125*x^5/5! - 1320*x^6/6! - 18249*x^7/7! - 290976*x^8/8! - 5385393*x^9/9! - 112642560*x^10/10! + ...
Also,
(A(x) - 1)/x  =  1 + x + 3*x^2/2! + 15*x^3/3! + 109*x^4/4! + 1040*x^5/5! + 12357*x^6/6! + 175980*x^7/7! + 2925345*x^8/8! + 55633824*x^9/9! + 1191857895*x^10/10! + 28409673600*x^11/11! + 745921397589*x^12/12! + ...
appears commonly in formulas for e.g.f. A(x).

Links

Crossrefs

Cf. A306066, A306090, A306067.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (-x)^m/m!*prod(k=0, m, (m+1-k) - (k+1)*Ser(A) ) ), #A-1)); n!*A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k) - (k+1)*A(x) = 0.
(2) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k - m) - (k+1)*A(x) = -m * A(x)^(2*m-2) * x^(m-2) / (A(x) - 1)^(m-2).
(3) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k) - (k+1 - p)*A(x) = p/A(x) * x^(p-2) / (A(x) - 1)^(p-2).
(4) Sum_{n>=0} (-x)^n/n! * Product_{k=0..n} (n+1-k - m) - (k+1 - p)*A(x) = -(m -p*A(x)) * A(x)^(2*m-2) * x^(m+p-2) / (A(x) - 1)^(m+p-2).
(5) A(x) = 1 / A(-x*A(x)).
(6) Sum_{n>=0} (-x)^n/n! * Product_{k=1..n} (n-k) - k*A(x) = A(x).
(7) Sum_{n>=0} (-x)^n/n! * Product_{k=1..n} (n-k+1) - (k-1)*A(x) = 1/A(x).
(8) Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1) - (2*k-1)*A(x) = 1.
(9) Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1 - m) - (2*k-1)*A(x) = A(x)^( m/(1 + A(x)) ).
(10) Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1) - (2*k-1 - p)*A(x) = A(x)^( -p*A(x)/(1 + A(x)) ).
(11) Sum_{n>=0} (-x/2)^n/n! * Product_{k=1..n} (2*(n-k)+1 - m) - (2*k-1 - p)*A(x) = A(x)^( (m - p*A(x))/(1 + A(x)) ).
a(n)/n! ~ c * d^n / n^(3/2), where d = 2.4559812888215554548... and c = 0.65281176845553367... - Vaclav Kotesovec, Jul 12 2018

A306092 G.f. A(x) satisfies: (1 + A(x))^A(x) = (1+x)^x, where A(x) = Sum_{n>=1} a(n)*x^n/(2*n-1)!.

Original entry on oeis.org

-1, 3, -30, 840, -45360, 3963960, -512431920, 91708016400, -21708518832000, 6566197230552960, -2470377569057798400, 1131411784221938419200, -619741850665486348800000, 400063411654998957081216000, -300571110264723992167009536000, 260020540519396684696076728320000, -256606704941070116606793272893440000, 286541492507208304817420296882114560000
Offset: 1

Views

Author

Paul D. Hanna, Jun 22 2018

Keywords

Comments

The g.f. is also given by: A(x) = Sum_{n>=0} A306090(n)/A306091(n) * x^n.

Examples

			G.f.: A(x) = -x + 3*x^2/3! - 30*x^3/5! + 840*x^4/7! - 45360*x^5/9! + 3963960*x^6/11! - 512431920*x^7/13! + 91708016400*x^8/15! - 21708518832000*x^9/17! + 6566197230552960*x^10/19! - 2470377569057798400*x^11/21! + 1131411784221938419200*x^12/23! - 619741850665486348800000*x^13/25! + ...
such that
(E.1) 1  =  1  +  (x + A(x))  +  (x + 2*A(x))*(2*x + A(x))/2!  +  (x + 3*A(x))*(2*x + 2*A(x))*(3*x + A(x))/3!  +  (x + 4*A(x))*(2*x + 3*A(x))*(3*x + 2*A(x))*(4*x + A(x))/4!  +  (x + 5*A(x))*(2*x + 4*A(x))*(3*x + 3*A(x))*(4*x + 2*A(x))*(5*x + A(x))/5! + ...
(E.2) (1 + x)^p  =  1  +  (x + (1-p)*A(x))  +  (x + (2-p)*A(x))*(2*x + (1-p)*A(x))/2!  +  (x + (3-p)*A(x))*(2*x + (2-p)*A(x))*(3*x + (1-p)*A(x))/3!  +  (x + (4-p)*A(x))*(2*x + (3-p)*A(x))*(3*x + (2-p)*A(x))*(4*x + (1-p)*A(x))/4! + ...
(E.3) (1 + A(x))^m  =  1  +  ((1-m)*x + A(x))  +  ((1-m)*x + 2*A(x))*((2-m)*x + A(x))/2!  +  ((1-m)*x + 3*A(x))*((2-m)*x + 2*A(x))*((3-m)*x + A(x))/3!  +  ((1-m)*x + 4*A(x))*((2-m)*x + 3*A(x))*((3-m)*x + 2*A(x))*((4-m)*x + A(x))/4! + ...
FUNCTIONAL EQUATION.
The series A(x) satisfies:
(E.4) (1 + A(x))^A(x) = (1 + x)^x  =  1 + x^2 - 1/2*x^3 + 5/6*x^4 - 3/4*x^5 + 33/40*x^6 - 5/6*x^7 + 2159/2520*x^8 - 209/240*x^9 + ...
GENERATING METHOD.
Although the functional equation (1 + A(x))^A(x) = (1 + x)^x has an infinite number of solutions, one may arrive at the g.f. A(x) by the following iteration.
If we start with A = -x, and iterate
(E.5) A = (A + x*log(1 + x)/log(1 + A))/2
then A will converge to g.f. A(x).

Links

Crossrefs

Cf. A306090, A306091.

Programs

  • PARI
    /* From Functional Equation (1 + A(x))^A(x) = (1 + x)^x */
{a(n) = my(A = -x +x*O(x^n)); for(i=1, n, A = (A + x*log(1+x +x*O(x^n))/log(1+A))/2 ); (2*n-1)! * polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n/(2*n-1)! satisfies:
(1) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + k*A(x) = 1.
(2) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k)*x + (k - p)*A(x) = (1 + x)^p.
(3) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + k*A(x) = (1 + A(x))^m.
(4) Sum_{n>=0} 1/n! * Product_{k=1..n} (n+1-k - m)*x + (k - p)*A(x) = (1+x)^p * (1 + A(x))^m.
(5) A(A(x)) = x.
(6) (1 + A(x))^A(x) = (1 + x)^x.

A306087 G.f. A(x) satisfies: Sum_{n>=0} Product_{k=1..n} (n+1-k)*x + k*A(x) = 1.

Original entry on oeis.org

-1, 1, -1, -2, 8, 56, -281, -3061, 18612, 271129, -1925781, -34967550, 284063311, 6174304311, -56535769915, -1431894779510, 14610783773266, 422789237646634, -4761801655073506, -155050750819877478, 1911855043475987609, 69202778917256845631, -927610459464373932427, -36955258706329671973028, 535191096878546873823897, 23273612576939618406997055, -362206459402896340382856127
Offset: 1

Views

Author

Paul D. Hanna, Jun 23 2018

Keywords

Examples

			G.f.: A(x) = -x + x^2 - x^3 - 2*x^4 + 8*x^5 + 56*x^6 - 281*x^7 - 3061*x^8 + 18612*x^9 + 271129*x^10 - 1925781*x^11 - 34967550*x^12 + 284063311*x^13 + 6174304311*x^14 - 56535769915*x^15 - 1431894779510*x^16 + ...
such that
1  =  1  +  (x + A(x))  +  (x + 2*A(x))*(2*x + A(x))  +  (x + 3*A(x))*(2*x + 2*A(x))*(3*x + A(x))  +  (x + 4*A(x))*(2*x + 3*A(x))*(3*x + 2*A(x))*(4*x + A(x))  +  (x + 5*A(x))*(2*x + 4*A(x))*(3*x + 3*A(x))*(4*x + 2*A(x))*(5*x + A(x)) + ...
also, A(A(x)) = x.

Links

Crossrefs

Cf. A306090.

Programs

  • PARI
    {a(n) = my(A=[-1]); for(i=1,n, A = concat(A,0); A[#A] = -Vec( sum(n=0,#A, prod(k=1,n, (n+1-k)*x + (k)*x*Ser(A) ) ) )[#A+1] );A[n]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) satisfies: A(A(x)) = x.
Showing 1-8 of 8 results.

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