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.

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A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.

Original entry on oeis.org

1, 2, 18, 260, 4890, 110124, 2844772, 82196424, 2613699450, 90450874860, 3379153837180, 135445714293720, 5796441493971284, 263784018974675416, 12721572505160772840, 648250134428292640272, 34809708051186914034730, 1965040180185473309749788, 116359823755204505172646204
Offset: 0

Views

Author

Paul D. Hanna, Feb 27 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 260*x^3 + 4890*x^4 + 110124*x^5 + 2844772*x^6 + 82196424*x^7 + 2613699450*x^8 + ... + a(n)*x^n + ...
where
A(x) = 1 + (d/dx x^2*A(x)^2) + (d^2/dx^2 x^4*A(x)^4)/2! + (d^3/dx^3 x^6*A(x)^6)/3! + (d^4/dx^4 x^8*A(x)^8)/4! + (d^5/dx^5 x^10*A(x)^10)/5! + (d^6/dx^6 x^12*A(x)^12)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(2*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)^2), which begins
B(x) = x + x^2 + 6*x^3 + 65*x^4 + 978*x^5 + 18354*x^6 + 406396*x^7 + 10274553*x^8 + 290411050*x^9 + ... + A360977(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( x*A(x)^2 + (d/dx x^3*A(x)^4)/2! + (d^2/dx^2 x^5*A(x)^6)/3! + (d^3/dx^3 x^7*A(x)^8)/4! + (d^4/dx^4 x^9*A(x)^10)/5! + (d^5/dx^5 x^11*A(x)^12)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(2*n))/n! + ... ).

Links

Crossrefs

Cf. A360950, A360973, A360975, A360977.
Cf. A229619, A356848, A088714, A303063.

Programs

  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^(2*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^2*A^2 +O(x^(n+2))))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^2).
(3) B(x - x^2*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(2*n) / n! ) is the g.f. of A360977.
(4) a(n) = (n+1) * A360977(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1/2)^n, where alfa = 2.498459235192... and c = 0.0920029178453... - Vaclav Kotesovec, Feb 28 2023
alfa = 5*LambertW(1/2) + 1/(1 + LambertW(1/2)). - Vaclav Kotesovec, Mar 13 2023

A067145 Shifts left under reversion.

Original entry on oeis.org

1, 1, -1, 3, -13, 69, -419, 2809, -20353, 157199, -1281993, 10963825, -97828031, 907177801, -8716049417, 86553001779, -886573220093, 9351927111901, -101447092428243, 1130357986741545, -12923637003161409, 151479552582252239, -1818756036793636033
Offset: 1

Views

Author

Christian G. Bower, Jan 03 2002

Keywords

Links

Crossrefs

Cf. A107094.
Apart from signs, same as A088714. - Philippe Deléham, Jun 18 2006
Cf. A309254, A309564.

Programs

  • Mathematica
    Nest[InverseSeries[#] x + x &, x + O[x]^2, 50][[3]] (* Vladimir Reshetnikov, Aug 07 2019 *)
  • Maxima
    T(n,m):=if n=m then 1 else m/n*sum(T(n-m,k)*(-1)^k*binomial(k+n-1,n-1), k,1,n-m); a(n):=if n=1 then 1 else T(n-1,1); /* Vladimir Kruchinin, May 06 2012 */
  • PARI
    {a(n)=local(A); if(n<1, 0, A=x+O(x^2); for(i=2,n, A=x*(1+serreverse(A))); polcoeff(A,n))} /* Michael Somos, May 21 2005 */

Formula

G.f. satisfies A^(-1)(x) = A(x)/x - 1.
G.f. satisfies: A(A(x)) = (1+x)*A(x) = g.f. of A107094. - Paul D. Hanna, May 12 2005
G.f. A(x) satisfies 0=f(x, A(x), A(A(x))) where f(a0,a1,a2) = a1 - a2 + a0*a1. - Michael Somos, May 21 2005
a(n) = T(n-1,1), n > 1, a(1) = 1, T(n,m) = (m/n) * Sum_{k=1..n-m} T(n-m,k) * (-1)^k * binomial(k+n-1, n-1), n > m, T(n,n) = 1. - Vladimir Kruchinin, May 06 2012

A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.

Original entry on oeis.org

1, 3, 45, 1113, 36459, 1448568, 66726309, 3469988835, 200242815669, 12670449226269, 871389659249424, 64693985439491127, 5156607707368927875, 439261264283443326927, 39831856169938193953827, 3831650468281643037364389, 389807188331526942149375433
Offset: 0

Views

Author

Paul D. Hanna, Mar 03 2023

Keywords

Examples

			G.f.: A(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + a(n)*x^(2*n) + ...
where
A(x) = 1 + (d/dx x^3*A(x)^2) + (d^2/dx^2 x^6*A(x)^4)/2! + (d^3/dx^3 x^9*A(x)^6)/3! + (d^4/dx^4 x^12*A(x)^8)/4! + (d^5/dx^5 x^15*A(x)^10)/5! + ... + (d^n/dx^n x^(3*n)*A(x)^(2*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)^2), which begins
B(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + 11778989157*x^17 + ... + A361047(n)*x^(2*n-1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^2*A(x)^2 + (d/dx x^5*A(x)^4)/2! + (d^2/dx^2 x^8*A(x)^6)/3! + (d^3/dx^3 x^11*A(x)^8)/4! + (d^4/dx^4 x^14*A(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^(2*n))/n! + ... ).

Links

Crossrefs

Cf. A361047, A360950, A360973, A360974, A360975, A360976.
Cf. A229619, A356848, A088714, A303063.

Programs

  • Mathematica
    nt = 40; (* number of terms to produce *)
A[_] = 0;
Do[A[x_] = D[InverseSeries[x - x^3*A[x]^2 + O[x]^k] // Normal, x], {k, 1, 2*nt}];
CoefficientList[A[x^(1/2)], x] (* Jean-François Alcover, Mar 04 2023 *)
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^(2*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^2 +O(x^(2*n+3))))); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^2).
(3) B(x - x^3*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(2*n) / n! ) is the g.f. of A361047.
(4) a(n) = (2*n+1) * A361047(n+1) for n >= 0.
a(n) == 0 (mod 3) for n > 0.
a(n) ~ c * 2^n * n! * n^alfa / LambertW(1/2)^n, where alfa = 1.623844426394406... and c = 0.18597481905555548924712403113114... - Vaclav Kotesovec, Mar 04 2023
alfa = (15*LambertW(1/2) - 1 + 3/(1 + LambertW(1/2)))/4. - Vaclav Kotesovec, Mar 15 2023

A088713 G.f. A(x) satisfies A(x/A(x)) = 1/(1-x).

Original entry on oeis.org

1, 1, 2, 6, 24, 118, 674, 4308, 30062, 225266, 1791964, 15009118, 131566314, 1201452248, 11389283418, 111761444078, 1132680800640, 11834071103246, 127261591139010, 1406778021294220, 15967144849210158, 185897394076705298
Offset: 0

Views

Author

Paul D. Hanna, Oct 12 2003

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 118*x^5 + 674*x^6 +...
Illustration of logarithmic derivation.
If we form an array of coefficients of x^k in A(x)^n, n>=1, like so:
A^1: [1],1,  2,   6,   24,   118,   674,    4308, ...;
A^2: [1, 2], 5,  16,   64,   308,  1716,   10724, ...;
A^3: [1, 3,  9], 31,  126,   600,  3278,   20070, ...;
A^4: [1, 4, 14,  52], 217,  1032,  5560,   33440, ...;
A^5: [1, 5, 20,  80,  345], 1651,  8820,   52270, ...;
A^6: [1, 6, 27, 116,  519,  2514],13385,   78420, ...;
A^7: [1, 7, 35, 161,  749,  3689, 19663], 114269, ...; ...
then the sums of the coefficients of x^k, k=0..n-1, in A(x)^n (shown above in brackets) begin:
1 = 1;
1 + 2 = 3;
1 + 3 +  9 = 13;
1 + 4 + 14 +  52 = 71;
1 + 5 + 20 +  80 +  345 = 451;
1 + 6 + 27 + 116 +  519 +  2514 = 3183;
1 + 7 + 35 + 161 +  749 +  3689 + 19663 = 24305; ...
and equal the coefficients in log(A(x)):
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 71*x^4/4 + 451*x^5/5 + 3183*x^6/6 + 24305*x^7/7 + 197551*x^8/8 +...
The main diagonal in the above table forms the g.f. G(x) of A088714:
[1/1, 2/2, 9/3, 52/4, 345/5, 2514/6, 19663/7, ...]
where G(x) = 1 + x + 3*x^2 + 13*x^3 + 69*x^4 + 419*x^5 + 2809*x^6 +...
satisfies A'(x)/A(x) = (G(x) + x*G'(x)) / (1 - x*G(x)).

Links

Crossrefs

Cf. A088714.
Variants: A154677, A168448, A168449, A168478, A168479.

Programs

  • Mathematica
    terms = 22; A[] = 1; Do[A[x] = 1 + x*A[x]*A[1 - 1/A[x]] + O[x]^j // Normal, {j, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)
  • PARI
    a(n)=local(A=1+x);for(i=1,n,A=(1+A*serreverse(x/(A+x*O(x^n))))^1);polcoeff(A,n)
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Dec 06 2009
  • PARI
    {a(n)=local(A=1+x);if(n==0,1,for(i=1,n,
A=1+x*exp(sum(k=1,n-1,sum(j=0,k,polcoeff(A^k+x*O(x^j),j))*x^k/k)+x*O(x^n))));
polcoeff(A+x*O(x^n),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Dec 09 2013

Formula

G.f. satisfies: A(x) = 1 + x*A(x)*A(1-1/A(x)).
G.f.: A(x*g(x)) = g(x) = (1-1/A(x))/x where g(x) is the g.f. of A088714.
From Paul D. Hanna, Dec 06 2009: (Start)
G.f. satisfies: A(x) = 1 + A(x)*Series_Reversion(x/A(x)).
G.f. satisfies: A( (x/(1+x)) / A(x/(1+x)) ) = 1 + x.
(End)
Logarithmic derivative: given g.f. A(x), let G(x) = A(x*G(x)) be the g.f. of A088714, then A'(x)/A(x) = (G(x) + x*G'(x)) / (1 - x*G(x)).

A212910 G.f. satisfies: A(x) = x^2 - x + Series_Reversion(x - x*A(x)).

Original entry on oeis.org

1, 1, 1, 4, 11, 35, 125, 445, 1699, 6668, 26935, 112111, 476674, 2072146, 9182141, 41406119, 189830984, 883549848, 4171598085, 19962224926, 96746007976, 474586282085, 2355104582435, 11817111373152, 59928222117495, 307045555880793, 1588825668984517
Offset: 2

Views

Author

Paul D. Hanna, May 30 2012

Keywords

Comments

Compare the g.f. to a g.f. G(x) of A088714 (offset 1), which satisfies:
G(x) = Series_Reversion(x - x*G(x)),
G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*G(x)^n/n!, and
G(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*G(x)^n/n! ).

Examples

			G.f.: A(x) = x^2 + x^3 + x^4 + 4*x^5 + 11*x^6 + 35*x^7 + 125*x^8 +...
The series reversion of x - x*A(x) begins:
x + x^3 + x^4 + 4*x^5 + 11*x^6 + 35*x^7 + 125*x^8 + 445*x^9 +...
which equals x - x^2 + A(x).
The g.f. satisfies:
A(x) = x^2 + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
log(1-x + A(x)/x) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...
Related expansions:
d/dx x^2*A(x)^2/2! = 3*x^5 + 7*x^6 + 12*x^7 + 45*x^8 + 155*x^9 +...
d^2/dx^2 x^3*A(x)^3/3! = 12*x^7 + 45*x^8 + 110*x^9 + 418*x^10 +...
d^3/dx^3 x^4*A(x)^4/4! = 55*x^9 + 286*x^10 + 910*x^11 + 3640*x^12 +...
d^4/dx^4 x^5*A(x)^5/5! = 273*x^11 + 1820*x^12 + 7140*x^13 +...
...
d^(n-1)/dx^(n-1) x^n*A(x)^n/n! = A001764(n)*x^(2*n+1) +...

Links

Crossrefs

Cf. A088714, A212919, A212922, A212923.

Programs

  • PARI
    {a(n)=local(A=x^2);for(i=1,n,A=x^2-x+serreverse(x-x*A +x*O(x^n)));polcoeff(A,n)}
for(n=2,35,print1(a(n),", "))
  • PARI
    {Dx(n,F)=local(G=F);for(i=1,n,G=deriv(G));G}
{a(n)=local(A=x^2);for(i=1,n,A=x^2+sum(m=1,n,Dx(m-1,x^m*A^m/m!)+x*O(x^n)));polcoeff(A,n)}
for(n=2,35,print1(a(n),", "))
  • PARI
    {Dx(n,F)=local(G=F);for(i=1,n,G=deriv(G));G}
{a(n)=local(A=x^2);for(i=1,n,A=x^2-x+x*exp(sum(m=1,n,Dx(m-1,x^(m-1)*A^m/m!)+x*O(x^n))));polcoeff(A,n)}
for(n=2,35,print1(a(n),", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x^2 + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.
(2) A(x) = x^2 - x + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).

A212919 G.f. satisfies: A(x) = x^3 - x + Series_Reversion(x - x*A(x)).

Original entry on oeis.org

1, 1, 1, 1, 5, 14, 29, 73, 229, 671, 1840, 5415, 16983, 52547, 161420, 511039, 1655598, 5372395, 17527912, 58076084, 194676024, 656160449, 2227549164, 7635624954, 26380508479, 91696805060, 320866223000, 1130833326852, 4010720214072, 14306769257286
Offset: 3

Views

Author

Paul D. Hanna, May 31 2012

Keywords

Comments

Compare the g.f. to a g.f. G(x) of A088714 (offset 1), which satisfies:
G(x) = Series_Reversion(x - x*G(x)),
G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*G(x)^n/n!, and
G(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*G(x)^n/n! ).

Examples

			G.f.: A(x) = x^3 + x^4 + x^5 + x^6 + 5*x^7 + 14*x^8 + 29*x^9 + 73*x^10 +...
The series reversion of x - x*A(x) begins:
x + x^4 + x^5 + x^6 + 5*x^7 + 14*x^8 + 29*x^9 + 73*x^10 + 229*x^11 +...
which equals x - x^3 + A(x).
The g.f. satisfies:
A(x) = x^3 + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
log(1-x^2 + A(x)/x) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...
Related expansions:
d/dx x^2*A(x)^2/2! = 4*x^7 + 9*x^8 + 15*x^9 + 22*x^10 + 78*x^11 + 260*x^12 +...
d^2/dx^2 x^3*A(x)^3/3! = 22*x^10 + 78*x^11 + 182*x^12 + 350*x^13 + 1080*x^14 +...
d^3/dx^3 x^4*A(x)^4/4! = 140*x^13 + 680*x^14 + 2040*x^15 + 4845*x^16 +...
d^4/dx^4 x^5*A(x)^5/5! = 969*x^16 + 5985*x^17 + 21945*x^18 + 61985*x^19 +...
...
d^(n-1)/dx^(n-1) x^n*A(x)^n/n! = A002293(n)*x^(3*n+1) +...

Links

Crossrefs

Cf. A088714, A212910.

Programs

  • PARI
    {a(n)=local(A=x^3); for(i=1, n, A=x^3-x+serreverse(x-x*A +x*O(x^n))); polcoeff(A, n)}
for(n=3, 40, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x^3); for(i=1, n, A=x^3+sum(m=1, n, Dx(m-1, x^m*A^m/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=3, 40, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{a(n)=local(A=x^3); for(i=1, n, A=x^3-x+x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*A^m/m!)+x*O(x^n)))); polcoeff(A, n)}
for(n=3, 40, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x^3 + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.
(2) A(x) = x^3 - x + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).

A140094 G.f. satisfies: A(x) = x/(1 - A(A(A(x)))).

Original entry on oeis.org

1, 1, 4, 25, 199, 1855, 19387, 221407, 2717782, 35455981, 487672243, 7029980797, 105732907498, 1653377947393, 26805765569863, 449568735630517, 7785116448484318, 138980739891821269, 2554369130466577138
Offset: 1

Views

Author

Paul D. Hanna, May 08 2008, May 20 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 199*x^5 + 1855*x^6 + 19387*x^7 +...
Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
A_2(x) = 1 - Series_Reversion( A(x) ) / x;
A_3(x) = 1 - x / A(x);
A_4(x) = 1 - A(x) / A_2(x);
A_5(x) = 1 - A_2(x) / A_3(x);
A_6(x) = 1 - A_3(x) / A_4(x); ...
where the iterations of A(x) begin:
A_2(x) = x + 2*x^2 + 10*x^3 + 71*x^4 + 616*x^5 + 6119*x^6 + 67210*x^7 +...;
A_3(x) = x + 3*x^2 + 18*x^3 + 144*x^4 + 1365*x^5 + 14544*x^6 +...;
A_4(x) = x + 4*x^2 + 28*x^3 + 250*x^4 + 2584*x^5 + 29584*x^6 +...;
A_5(x) = x + 5*x^2 + 40*x^3 + 395*x^4 + 4435*x^5 + 54515*x^6 +...;
A_6(x) = x + 6*x^2 + 54*x^3 + 585*x^4 + 7104*x^5 + 93555*x^6 +...;
...
Iterations are also related by continued fractions:
A(x) = x/(1 - A_2(x)/(1 - A_4(x)/(1 - A_6(x)/(1 -...)))) ;
A_2(x) = A(x)/(1 - A_3(x)/(1 - A_5(x)/(1 - A_7(x)/(1 -...)))).

Links

Crossrefs

Cf. A140095, A088714.
Cf. A088717, A091713, A120971, A139702.

Programs

  • PARI
    {a(n)=local(A); if(n<0, 0, n++; A=x+O(x^2); for(i=2, n, A=x/(1-subst(A, x, subst(A, x, A)))); polcoeff(A, n))}

Formula

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion(x - x*A(A(x))).
(2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n / n!.
(3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(x))^n/x / n! ).
Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
then A_{n}(x) = A_{n-1}/[1 - A_{n+2}(x)] ;
thus A_{n}(x) = 1 - A_{n-3}(x) / A_{n-2}(x).
G.f. A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 + x*A*C;
B = A + x*B*D;
C = B + x*C*E;
D = C + x*D*F;
E = D + x*E*G; ...

Extensions

Name, formulas, and examples revised by Paul D. Hanna, Feb 03 2013

A140095 G.f. satisfies: A(x) = x/(1 - A(A(A(A(x))))).

Original entry on oeis.org

1, 1, 5, 41, 437, 5513, 78477, 1225865, 20644021, 370334137, 7017055933, 139562915193, 2899946191077, 62722686552841, 1408033260333581, 32729098457253417, 786224322656857941, 19486950945070339801, 497649167866430159197, 13078602790892074110937
Offset: 1

Views

Author

Paul D. Hanna, May 08 2008, May 20 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 41*x^4 + 437*x^5 + 5513*x^6 + 78477*x^7 +...
Iterations A_{n+1}(x) = A( A_{n}(x) ) are related as follows.
A_2(x) = 1 - Series_Reversion(A_2(x)) / Series_Reversion(A(x));
A_3(x) = 1 - Series_Reversion(A(x)) / x;
A_4(x) = 1 - x / A(x);
A_5(x) = 1 - A(x) / A_2(x);
A_6(x) = 1 - A_2(x) / A_3(x);
A_7(x) = 1 - A_3(x) / A_4(x);
A_8(x) = 1 - A_4(x) / A_5(x); ...
where the iterations of A(x) begin:
A_2(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1220*x^5 + 16028*x^6 +...
A_3(x) = x + 3*x^2 + 21*x^3 + 207*x^4 + 2489*x^5 + 34259*x^6 +...
A_4(x) = x + 4*x^2 + 32*x^3 + 344*x^4 + 4408*x^5 + 63776*x^6 +...
A_5(x) = x + 5*x^2 + 45*x^3 + 525*x^4 + 7165*x^5 + 109125*x^6 +...
A_6(x) = x + 6*x^2 + 60*x^3 + 756*x^4 + 10972*x^5 + 175948*x^6 +...
A_7(x) = x + 7*x^2 + 77*x^3 + 1043*x^4 + 16065*x^5 + 271103*x^6 +...
A_8(x) = x + 8*x^2 + 96*x^3 + 1392*x^4 + 22704*x^5 + 402784*x^6 +...
...
Iterations are also related by continued fractions:
A(x) = x/(1 - A_3(x)/(1 - A_6(x)/(1 - A_9(x)/(1 -...)))) ;
A_2(x) = A(x)/(1 - A_4(x)/(1 - A_7(x)/(1 - A_10(x)/(1 -...)))) ;
A_3(x) = A_2(x)/(1 - A_5(x)/(1 - A_8(x)/(1 - A_11(x)/(1 -...)))) ;
A_4(x) = A_3(x)/(1 - A_6(x)/(1 - A_9(x)/(1 - A_12(x)/(1 -...)))) ; ...

Crossrefs

Cf. A140094, A088714, A088717, A091713, A120971, A139702.

Programs

  • PARI
    {a(n)=local(A); if(n<1, 0, n++; A=x+O(x^2); for(i=2, n,B=subst(A, x, A); A=x/(1-subst(B, x, B))); polcoeff(A, n))}

Formula

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion(x - x*A(A(A(x)))).
(2) A(x) = x + Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(A(x)))^n / n!.
(3) A(x) = x*exp( Sum_{n=1} d^(n-1)/dx^(n-1) x^n * A(A(A(x)))^n/x / n! ).
Define A_{n} such that A_{n+1}(x) = A( A_{n}(x) ) with A_0(x) = x,
then A_{n}(x) = A_{n-1}/[1 - A_{n+3}(x)] ;
thus A_{n}(x) = 1 - A_{n-4}(x) / A_{n-3}(x).
G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 + x*A*D;
B = A + x*B*E;
C = B + x*C*F;
D = C + x*D*G;
E = D + x*E*H; ...

A212922 G.f. satisfies: A(x) = x^2/(1-x) + Series_Reversion(x - x*A(x)).

Original entry on oeis.org

1, 2, 5, 21, 120, 800, 5881, 46565, 391876, 3473879, 32226510, 311313683, 3119693862, 32333294383, 345754479372, 3807294710182, 43101806735623, 500977869387150, 5971566838065819, 72925079326977943, 911614856156206061, 11656341547670071145, 152347288068103795503
Offset: 1

Views

Author

Paul D. Hanna, May 31 2012

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 5*x^3 + 21*x^4 + 120*x^5 + 800*x^6 + 5881*x^7 +...
The series reversion of x - x*A(x) begins:
x + x^2 + 4*x^3 + 20*x^4 + 119*x^5 + 799*x^6 + 5880*x^7 +...
which equals A(x) - x^2/(1-x).
The g.f. A(x) satisfies:
A(x) - x^2/(1-x) = x + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
log(A(x)/x - x/(1-x)) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...

Links

Crossrefs

Cf. A212923, A088714, A212910, A212919.

Programs

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

Formula

G.f. A(x) also satisfies:
(1) A(x) = x/(1-x) + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.
(2) A(x) = x^2/(1-x) + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).

A212923 G.f. satisfies: A(x) = x^2 + Series_Reversion(x - x*A(x)).

Original entry on oeis.org

1, 2, 4, 19, 111, 734, 5338, 41839, 348827, 3065255, 28199803, 270253498, 2687629926, 27652068276, 293627150268, 3211604669731, 36124424800797, 417294625090201, 4944772338009206, 60045368928594948, 746560751627818906, 9496624640844863631, 123507266690219103213
Offset: 1

Views

Author

Paul D. Hanna, May 31 2012

Keywords

Comments

This is an application of the more general formula given by:
if G(x) = Series_Reversion(x - x*F(x)), with F(0)=0, then
(1) G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*F(x)^n/n!,
(2) G(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*F(x)^n/n! );
here F(x) = A(x) and G(x) = A(x) - x^2.

Examples

			G.f.: A(x) = x + 2*x^2 + 4*x^3 + 19*x^4 + 111*x^5 + 734*x^6 + 5338*x^7 +...
The series reversion of x - x*A(x) begins:
x + x^2 + 4*x^3 + 19*x^4 + 111*x^5 + 734*x^6 + 5338*x^7 +...
which equals A(x) - x^2.
The g.f. A(x) satisfies:
A(x) - x^2 = x + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
log(A(x)/x - x) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...

Links

Crossrefs

Cf. A212922, A088714, A212910, A212919.

Programs

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

Formula

G.f. A(x) also satisfies:
(1) A(x) = x+x^2 + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.
(2) A(x) = x^2 + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).
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