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 1-6 of 6 results.

A210949 E.g.f. A(x) satisfies: A'(x) = 1/(1 - A(A(x))).

Original entry on oeis.org

1, 1, 4, 29, 309, 4383, 78121, 1684706, 42801222, 1255919755, 41918624013, 1572257236114, 65619165625383, 3022617826829288, 152615633802149416, 8397224009015443509, 500957609480739613321, 32261529179806961067634, 2234133327582388824135291
Offset: 1

Views

Author

Paul D. Hanna, Jul 22 2012

Keywords

Comments

An unsigned version of A067146.
Equals row sums of triangle A277410.
Is this a duplicate of A014622, which is related to f(f(x))f'(x)=-1 ? - R. J. Mathar, May 13 2025

Examples

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 29*x^4/4! + 309*x^5/5! + 4383*x^6/6! +...
Let G(x) = Integral A(x) dx, then A(x) = x + G(A(x)) where
G(x) = x^2/2! + x^3/3! + 4*x^4/4! + 29*x^5/5! + 309*x^6/6! + 4383*x^7/7! +...
Also,
A(x) = x + G(x) + d/dx G(x)^2/2! + d^2/dx^2 G(x)^3/3! + d^3/dx^3 G(x)^4/4! +...
log(A(x)/x) = G(x)/x + d/dx G(x)^2/(2!*x) + d^2/dx^2 G(x)^3/(3!*x) + d^3/dx^3 G(x)^4/(4!*x) +...
By definition, A'(x) = 1/(1 - A(A(x))), where
A(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 101*x^4/4! + 1313*x^5/5! + 22235*x^6/6! + 466356*x^7/7! + 11710760*x^8/8! +...

Links

Crossrefs

Cf. A277410, A067146, A279843, A279844, A279845, A280570, A280571, A280572, A280573, A280574, A280575.

Programs

  • Mathematica
    m = 20; A[_] = 0;
Do[A[x_] = InverseSeries[Integrate[1 - A[x], x] + O[x]^m], {m}];
CoefficientList[A[x], x] * Range[0, m - 1]! // Rest (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n)=local(A=x,G);for(i=1,n,G=intformal(A+x*O(x^n));A=serreverse(x-G));n!*polcoeff(A, n)}
  • PARI
    {a(n)=local(A=x,G);for(i=1,n,G=intformal(A+x*O(x^n));A=x+subst(G,x,A+x*O(x^n))); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x,G);for(i=0,n,G=intformal(A+x*O(x^n)); A=x+sum(m=1, n, Dx(m-1, G^m/m!))); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x,G);for(i=0,n,G=intformal(A+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, G^m/x/m!)+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {a(n) = local(A=x); for(i=1,n, A = serreverse(intformal(1-A +x*O(x^n)))); n!*polcoeff(A,n)}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral 1 - A(x) dx ).
(2) A''(x) = 1 / ( (1 - A(A(x)))^3 * (1 - A(A(A(x)))) ).
Let G(x) = Integral A(x) dx with G(0)=0, then the e.g.f. A(x) satisfies:
(3) A(x) = x + G(A(x)) or, equivalently, A(x - G(x)) = x.
(4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) G(x)^n / n!.
(5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) G(x)^n/(n!*x) ).
a(n) = Sum_{k=0..n-1} A277410(n,k).

A277410 G.f. A(x,y) satisfies: A( x - y*G(x,y), y) = x + (1-y)*G(x,y) such that G(x,y) = Integral A(x,y) dx, where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

Original entry on oeis.org

1, 1, 0, 1, 3, 0, 1, 13, 15, 0, 1, 38, 165, 105, 0, 1, 94, 1033, 2310, 945, 0, 1, 213, 4953, 26229, 36330, 10395, 0, 1, 459, 20370, 213511, 674520, 640710, 135135, 0, 1, 960, 76056, 1421225, 8559675, 18127935, 12588345, 2027025, 0, 1, 1972, 266334, 8283234, 85654979, 337805535, 515903850, 273544425, 34459425, 0, 1, 4007, 892542, 44013478, 729292193, 4822487682, 13506364410, 15631793100, 6529047525, 654729075, 0
Offset: 1

Views

Author

Paul D. Hanna, Oct 13 2016

Keywords

Comments

More generally, we have the following related identity.
Given functions F and G with F(0)=0, F'(0)=1, G(0)=0, G'(0)=0,
if F(x - y*G(x)) = x + (1-y)*G(x), then
(1) F(x) = x + G( y*F(x) + (1-y)*x ),
(2) y*F(x) + (1-y)*x = Series_Reversion(x - y*G(x)),
(3) F(x) = x + G(x + y*G(x + y*G(x + y*G(x +...)))),
(4) F(x) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x)^n / n!.
The g.f. of this sequence A(x,y) equals F(x) in the above when G(x) = Integral F(x) dx.

Examples

			G.f.: A(x,y) = x + x^2/2! + (3*y + 1)*x^3/3! + (15*y^2 + 13*y + 1)*x^4/4! + (105*y^3 + 165*y^2 + 38*y + 1)*x^5/5! + (945*y^4 + 2310*y^3 + 1033*y^2 + 94*y+ 1)*x^6/6! + (10395*y^5 + 36330*y^4 + 26229*y^3 + 4953*y^2 + 213*y + 1)*x^7/7! + (135135*y^6 + 640710*y^5 + 674520*y^4 + 213511*y^3 + 20370*y^2 + 459*y + 1)*x^8/8! + (2027025*y^7 + 12588345*y^6 + 18127935*y^5 + 8559675*y^4 + 1421225*y^3 + 76056*y^2 + 960*y + 1)*x^9/9! + (34459425*y^8 + 273544425*y^7 + 515903850*y^6 + 337805535*y^5 + 85654979*y^4 + 8283234*y^3 + 266334*y^2 + 1972*y + 1)*x^10/10! +...
such that A( x - y*G(x,y), y)  =  x + (1-y)*G(x,y)
also,
A(x,y) = x + G( y*A(x,y) + (1-y)*x, y)
where G(x,y) = Integral A(x,y).
...
This triangle of coefficients T(n,k) of x^n*y^k/n! in g.f. A(x,y) begins:
1;
1, 0;
1, 3, 0;
1, 13, 15, 0;
1, 38, 165, 105, 0;
1, 94, 1033, 2310, 945, 0;
1, 213, 4953, 26229, 36330, 10395, 0;
1, 459, 20370, 213511, 674520, 640710, 135135, 0;
1, 960, 76056, 1421225, 8559675, 18127935, 12588345, 2027025, 0;
1, 1972, 266334, 8283234, 85654979, 337805535, 515903850, 273544425, 34459425, 0;
1, 4007, 892542, 44013478, 729292193, 4822487682, 13506364410, 15631793100, 6529047525, 654729075, 0;
1, 8089, 2900353, 218797958, 5531376285, 57226590953, 264482764305, 555756298020, 505173143475, 170116046100, 13749310575, 0; ...
in which the diagonal equals A001147 (odd double factorials), and the row sums yield A210949.
...
APPLICATION.
Given F(x) such that
F(x - Integral p*F(x) dx) = x + Integral q*F(x) dx
then
F(x) = Sum_{n>=1} a(n)*x^n/n!
where
a(n) = Sum_{k=0..n-1} A277410(n,k) * p^k * (p+q)^(n-k-1) for n>=1.
EXAMPLES.
A210949(n) = Sum_{k=0..n-1} A277410(n,k).
A277403(n) = Sum_{k=0..n-1} A277410(n,k) * 2^(n-k-1).
A279843(n) = Sum_{k=0..n-1} A277410(n,k) * 3^(n-k-1).
A279844(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 3^(n-k-1).
A279845(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k.
A280570(n) = Sum_{k=0..n-1} A277410(n,k) * 4^(n-k-1).
A280571(n) = Sum_{k=0..n-1} A277410(n,k) * 3^k * 4^(n-k-1).
A280572(n) = Sum_{k=0..n-1} A277410(n,k) * 5^(n-k-1).
A280573(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 5^(n-k-1).
A280574(n) = Sum_{k=0..n-1} A277410(n,k) * 3^k * 5^(n-k-1).
A280575(n) = Sum_{k=0..n-1} A277410(n,k) * 4^k * 5^(n-k-1).
...
COLUMN GENERATING FUNCTIONS.
From _Paul D. Hanna_, Nov 05 2016: (Start)
_Colin Barker_ observed that column 1 of this triangle (A277411) appears to have the o.g.f. x*(3*x-2*x^2) / ((1-x)^3*(1-2*x)).
This observation led to the following conjecture.
Let F(k,x) = o.g.f. of column k in this triangle,
then
F(k,x) = P(k,x) * x^(k+1) / Product_{j=0..k} (1 - (j+1)*x)^(2*(k-j)+1)
where P(k,x) is a polynomial in x with degree k*(k+1) for k>=0.
Example:
F(0,x) = x/(1-x) ;
F(1,x) = P(1,x)*x^2/((1-x)^3*(1-2*x)) ;
F(2,x) = P(2,x)*x^3/((1-x)^5*(1-2*x)^3*(1-3*x)) ;
F(3,x) = P(3,x)*x^4/((1-x)^7*(1-2*x)^5*(1-3*x)^3*(1-4*x)) ;
...
The polynomials P(k,x) begin:
P(0,x) = 1 ;
P(1,x) = 3*x - 2*x^2 ;
P(2,x) = 15*x - 45*x^2 - 2*x^3 + 106*x^4 - 92*x^5 + 24*x^6 ;
P(3,x) = 105*x - 840*x^2 + 504*x^3 + 16321*x^4 - 75880*x^5 + 154483*x^6 - 152077*x^7 + 39208*x^8 + 59000*x^9 - 60336*x^10 + 23328*x^11 - 3456*x^12 ;
P(4,x) = 945*x - 15645*x^2 + 32445*x^3 + 1255770*x^4 - 15120061*x^5 + 86803308*x^6 - 291640845*x^7 + 529758178*x^8 - 50236668*x^9 - 2553002523*x^10 + 7695202852*x^11 - 12713196156*x^12 + 13351222596*x^13 - 8752472980*x^14 + 2871967920*x^15 + 387984096*x^16 - 884504448*x^17 + 427064832*x^18 - 100694016*x^19 + 9953280*x^20 ;
P(5,x) = 10395*x - 305235*x^2 + 1299375*x^3 + 77300220*x^4 - 1834009998*x^5 + 21447595316*x^6 - 156933684108*x^7 + 721294719700*x^8 - 1490891586137*x^9 - 5868653004882*x^10 + 70213320019895*x^11 - 359261247450016*x^12 + 1234731543184308*x^13 - 3081038591203028*x^14 + 5553265322783926*x^15 - 6518085613542516*x^16 + 2256970375232288*x^17 + 9498116639867573*x^18 - 25485484994020128*x^19 + 37162639109810884*x^20 - 37419816866322296*x^21 + 27200926921683600*x^22 - 14055671260790656*x^23 + 4698364855901568*x^24 - 583485067952640*x^25 - 341605998065664*x^26 + 237336648708096*x^27 - 72380729917440*x^28 + 11910492979200*x^29 - 859963392000*x^30 ;
...
where the coefficient of x^(k*(k+1)) in P(k,x) equals A059332(k+1).
(End)

Links

Crossrefs

Cf. A210949 (row sums), A067146, A001147 (diagonal), A277411 (column 1), A277412 (diagonal).
Cf. A279843, A279844, A279845, A280570, A280571, A280572, A280573, A280574, A280575.

Programs

  • PARI
    {T(n, k) = my(A=x); for(i=1, n, A = x + subst(intformal(A +x*O(x^n)), x, y*A + (1-y)*x ) ); n!*polcoeff(polcoeff(A,n,x),k,y)}
for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

Formula

Given g.f. A(x,y), define G(x,y) = Integral A(x,y) dx, then
(1) A(x,y) = x + G( y*A(x,y) + (1-y)*x, y),
(2) y*A(x,y) + (1-y)*x = Series_Reversion( x - y*G(x,y) ),
(3) y*x + (1-y)*B(x,y) = Series_Reversion( x + (1-y)*G(x,y) ), where B( A(x,y), y) = x.
(4) A(x,y) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x,y)^n / n!.
In formulas 2 and 3, the series reversion is taken with respect to variable x.

A279844 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x + Integral A(x) dx.

Original entry on oeis.org

1, 3, 27, 441, 10593, 338715, 13603923, 660689217, 37773985257, 2492351980659, 186888829248171, 15733456044557193, 1472423968474987185, 151932311464679521803, 17166519680224611739203, 2111435499783771418877073, 281279117575497421255121721, 40406056752677361995435879907, 6234806360224720540046684747547, 1029860015641146082486445487150681
Offset: 1

Views

Author

Paul D. Hanna, Dec 30 2016

Keywords

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 27*x^3/3! + 441*x^4/4! + 10593*x^5/5! + 338715*x^6/6! + 13603923*x^7/7! + 660689217*x^8/8! + 37773985257*x^9/9! + 2492351980659*x^10/10! + 186888829248171*x^11/11! + 15733456044557193*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 2*G(x)) = x + G(x) where
G(x) = x^2/2! + 3*x^3/3! + 27*x^4/4! + 441*x^5/5! + 10593*x^6/6! + 338715*x^7/7! + 13603923*x^8/8! + 660689217*x^9/9! + 37773985257*x^10/10! +...
Also, A(x) = x + 3 * G( (2*A(x) + x)/3 ).
RELATED SERIES.
We have (2*A(x) + x)/3 = Series_Reversion( x - Integral 2*A(x) dx ), where
(2*A(x) + x)/3 = x + 2*x^2/2! + 18*x^3/3! + 294*x^4/4! + 7062*x^5/5! + 225810*x^6/6! + 9069282*x^7/7! + 440459478*x^8/8! + 25182656838*x^9/9! + 1661567987106*x^10/10! + 124592552832114*x^11/11! + 10488970696371462*x^12/12! +...
Further, A( (2*A(x) + x)/3 ) = (A'(x) - 1)/(2*A'(x) + 1), which begins
A( (2*A(x) + x)/3 ) = x + 5*x^2/2! + 63*x^3/3! + 1311*x^4/4! + 38445*x^5/5! + 1464381*x^6/6! + 68939271*x^7/7! + 3879180855*x^8/8! + 254691006453*x^9/9! + 19160241768837*x^10/10! + 1628342402620383*x^11/11! + 154564849209408975*x^12/12! +...

Links

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279845, A280570, A280571, A280572, A280573, A280574, A280575.

Programs

  • Mathematica
    m = 21; A[_] = 0;
Do[G[x_] = Integrate[A[x], x]; A[x_] = x + 3 G[(2 A[x] + x)/3] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! // Rest (* Jean-François Alcover, Oct 20 2019 *)
  • PARI
    /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
{a(n, p=2, q=1) = my(A=x, G); for(i=1, n, G = intformal(A +x*O(x^n)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q)) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n, 2, 1), ", "))
  • PARI
    /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=2, q=1) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); G=intformal(F); A[#A] = -polcoeff(subst(F, x, x - p*G) - q*G, #A) ); n!*A[n]}
for(n=1, 30, print1(a(n, 2, 1), ", "))
  • PARI
    /* Informal code to generate the first N terms: */
{N=20; p=2; q=1; A=x; for(i=1, N, G=intformal(A +x*O(x^N)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q))); Vec(serlaplace(A))}

Formula

E.g.f. A(x) satisfies:
(1) A(x - Integral 2*A(x) dx) = x + Integral A(x) dx.
(2) A(x) = x + 3 * G( (2*A(x) + x)/3 ), where G(x) = Integral A(x) dx.
(3) A(x) = -x/2 + 3/2 * Series_Reversion(x - Integral 2*A(x) dx).
(4) A( (2*A(x) + x)/3 ) = (A'(x) - 1)/(2*A'(x) + 1).
(5) A'(x - Integral 2*A(x) dx) = (1 + A(x))/(1 - 2*A(x)).
a(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 3^(n-k-1).

A280571 E.g.f. satisfies: A(x - Integral 3*A(x) dx) = x + Integral A(x) dx.

Original entry on oeis.org

1, 4, 52, 1228, 42652, 1972324, 114581476, 8051020348, 666126945340, 63620722928308, 6907454641512244, 842227742687112604, 114192665828161184332, 17076069626235659815108, 2796969496541969481342100, 498871283058754285439126092, 96403472225110465517090352700, 20094942949266343527252229063204, 4500802213556155723422379457382916, 1079478060677848794106956676648220860
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2017

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 52*x^3/3! + 1228*x^4/4! + 42652*x^5/5! + 1972324*x^6/6! + 114581476*x^7/7! + 8051020348*x^8/8! + 666126945340*x^9/9! + 63620722928308*x^10/10! + 6907454641512244*x^11/11! + 842227742687112604*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 3*G(x)) = x + G(x) where
G(x) = x^2/2! + 4*x^3/3! + 52*x^4/4! + 1228*x^5/5! + 42652*x^6/6! + 1972324*x^7/7! + 114581476*x^8/8! + 8051020348*x^9/9! + 666126945340*x^10/10! + 63620722928308*x^11/11! + 6907454641512244*x^12/12! +...
Also, A(x) = x + 4 * G( (3*A(x) + x)/4 ).
RELATED SERIES.
We have (3*A(x) + x)/4 = Series_Reversion( x - Integral 3*A(x) dx ), where
(3*A(x) + x)/4 = x + 3*x^2/2! + 39*x^3/3! + 921*x^4/4! + 31989*x^5/5! + 1479243*x^6/6! + 85936107*x^7/7! + 6038265261*x^8/8! + 499595209005*x^9/9! + 47715542196231*x^10/10! + 5180590981134183*x^11/11! + 631670807015334453*x^12/12! +...
Further, A( (3*A(x) + x)/4 ) = (A'(x) - 1)/(3*A'(x) + 1), which begins
A( (3*A(x) + x)/4 ) = x + 7*x^2/2! + 127*x^3/3! + 3817*x^4/4! + 161881*x^5/5! + 8924923*x^6/6! + 608517595*x^7/7! + 49615007497*x^8/8! + 4722073055173*x^9/9! + 515139762620935*x^10/10! + 63506672456719651*x^11/11! + 8747178021763399909*x^12/12! +...

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280570, A280572, A280573, A280574, A280575.

Programs

  • Mathematica
    m = 21; A[_] = 0;
Do[A[x_] = -x/3 + 4/3 InverseSeries[x-Integrate[3 A[x], x] + O[x]^m], {m}];
CoefficientList[A[x], x]*Range[0, m-1]! // Rest (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
{a(n, p=3, q=1) = my(A=x, G); for(i=1, n, G = intformal(A +x*O(x^n)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q)) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n, 3, 1), ", "))
  • PARI
    /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=3, q=1) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); G=intformal(F); A[#A] = -polcoeff(subst(F, x, x - p*G) - q*G, #A) ); n!*A[n]}
for(n=1, 30, print1(a(n, 3, 1), ", "))
  • PARI
    /* Informal code to generate the first N terms: */
{N=20; p=3; q=1; A=x; for(i=1, N, G=intformal(A +x*O(x^N)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q))); Vec(serlaplace(A))}

Formula

E.g.f. A(x) satisfies:
(1) A(x - Integral 3*A(x) dx) = x + Integral A(x) dx.
(2) A(x) = x + 4 * G( (3*A(x) + x)/4 ), where G(x) = Integral 3*A(x) dx.
(3) A(x) = -x/3 + 4/3 * Series_Reversion(x - Integral 3*A(x) dx).
(4) A( (3*A(x) + x)/4 ) = (A'(x) - 1)/(3*A'(x) + 1).
(5) A'(x - Integral 3*A(x) dx) = (1 + A(x))/(1 - 3*A(x)).
a(n) = Sum_{k=0..n-1} A277410(n,k) * 3^k * 4^(n-k-1).

A280572 E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 4*A(x) dx.

Original entry on oeis.org

1, 5, 40, 525, 10025, 253475, 8015725, 305359050, 13645726250, 701304298375, 40822454374125, 2658840618527250, 191861336190647375, 15213199343853357500, 1316408013706224687500, 123576861126283832953125, 12521371849855149886590625, 1363361618975383978443843750, 158900334287408210286438971875, 19755940413686794723417400000000, 2612146114817877629253999384562500, 366294181903982533559997504649828125
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2017

Keywords

Examples

			E.g.f.: A(x) = x + 5*x^2/2! + 40*x^3/3! + 525*x^4/4! + 10025*x^5/5! + 253475*x^6/6! + 8015725*x^7/7! + 305359050*x^8/8! + 13645726250*x^9/9! + 701304298375*x^10/10! + 40822454374125*x^11/11! + 2658840618527250*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 4*G(x) where
G(x) = x^2/2! + 5*x^3/3! + 40*x^4/4! + 525*x^5/5! + 10025*x^6/6! + 253475*x^7/7! + 8015725*x^8/8! + 305359050*x^9/9! + 13645726250*x^10/10! + 701304298375*x^11/11! + 40822454374125*x^12/12! +...
Also, A(x) = x + 5 * G( (A(x) + 4*x)/5 ).
RELATED SERIES.
We have (A(x) + 4*x)/5 = Series_Reversion( x - Integral A(x) dx ), where
(A(x) + 4*x)/5 = x + x^2/2! + 8*x^3/3! + 105*x^4/4! + 2005*x^5/5! + 50695*x^6/6! + 1603145*x^7/7! + 61071810*x^8/8! + 2729145250*x^9/9! + 140260859675*x^10/10! + 8164490874825*x^11/11! + 531768123705450*x^12/12! +...
Further, A( (A(x) + 4*x)/5 ) = (A'(x) - 1)/(A'(x) + 4), which begins
A( (A(x) + 4*x)/5 ) = x + 6*x^2/2! + 63*x^3/3! + 1045*x^4/4! + 24105*x^5/5! + 716195*x^6/6! + 26137820*x^7/7! + 1134457060*x^8/8! + 57203895725*x^9/9! + 3292221321425*x^10/10! + 213282348138700*x^11/11! + 15380885339509825*x^12/12! +...

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280570, A280571, A280573, A280574, A280575.

Programs

  • Mathematica
    m = 23; A[_] = 0;
Do[A[x_] = -4 x + 5 InverseSeries[x - Integrate[A[x], x] + O[x]^m], {m}];
CoefficientList[A[x], x] * Range[0, m-1]! // Rest (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
{a(n, p=1, q=4) = my(A=x, G); for(i=1, n, G = intformal(A +x*O(x^n)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q)) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n, 1, 4), ", "))
  • PARI
    /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=1, q=4) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); G=intformal(F); A[#A] = -polcoeff(subst(F, x, x - p*G) - q*G, #A) ); n!*A[n]}
for(n=1, 30, print1(a(n, 1, 4), ", "))
  • PARI
    /* Informal code to generate the first N terms: */
{N=20; p=1; q=4; A=x; for(i=1, N, G=intformal(A +x*O(x^N)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q))); Vec(serlaplace(A))}

Formula

E.g.f. A(x) satisfies:
(1) A(x - Integral A(x) dx) = x + Integral 4*A(x) dx.
(2) A(x) = x + 5 * G( (A(x) + 4*x)/5 ), where G(x) = Integral A(x) dx.
(3) A(x) = -4*x + 5 * Series_Reversion(x - Integral A(x) dx).
(4) A( (A(x) + 4*x)/5 ) = (A'(x) - 1)/(A'(x) + 4).
(5) A'(x - Integral A(x) dx) = (1 + 4*A(x))/(1 - A(x)).
a(n) = Sum_{k=0..n-1} A277410(n,k) * 5^(n-k-1).

A279845 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x - Integral A(x) dx.

Original entry on oeis.org

1, 1, 7, 87, 1577, 37921, 1143991, 41734167, 1793837945, 89100737537, 5038278258759, 320488252355991, 22712229678267017, 1778818548078114337, 152926844472960316055, 14348332105800041202903, 1461880180517958608890585, 161034066043430013259095681, 19105043857756090069661974951, 2432865068875486088572762200535, 331511875063241457659846364208233, 48205214775404458968179455649349921, 7461345443274460130807423699070922103
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2017

Keywords

Examples

			E.g.f.: A(x) = x + x^2/2! + 7*x^3/3! + 87*x^4/4! + 1577*x^5/5! + 37921*x^6/6! + 1143991*x^7/7! + 41734167*x^8/8! + 1793837945*x^9/9! + 89100737537*x^10/10! + 5038278258759*x^11/11! + 320488252355991*x^12/12! + 22712229678267017*x^13/13! + 1778818548078114337*x^14/14! + 152926844472960316055*x^15/15! +...

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A280570, A280571, A280572, A280573, A280574, A280575.

Programs

  • PARI
    /* A(x) = x + (p+q)*G((p*A(x) + q*x)/(p+q)) ; G(x) = Integral A(x) dx: */
{a(n, p=2, q=-1) = my(A=x, G); for(i=1, n, G = intformal(A +x*O(x^n)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q)) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n, 2, -1), ", "))
  • PARI
    /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=2, q=-1) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); G=intformal(F); A[#A] = -polcoeff(subst(F, x, x - p*G) - q*G, #A) ); n!*A[n]}
for(n=1, 30, print1(a(n, 2, -1), ", "))
  • PARI
    /* Informal code to generate the first N terms: */
{N=20; p=2; q=-1; A=x; for(i=1, N, G=intformal(A +x*O(x^N)); A = x + (p+q)*subst(G, x, (p*A + q*x)/(p+q))); Vec(serlaplace(A))}

Formula

E.g.f. A(x) satisfies:
(1) A(x - Integral 2*A(x) dx) = x - Integral A(x) dx.
(2) A(x) = x + G( 2*A(x) - x ), where G(x) = Integral A(x) dx.
(3) A(x) = x/2 + 1/2 * Series_Reversion(x - Integral 2*A(x) dx).
(4) A( 2*A(x) - x ) = (A'(x) - 1)/(2*A'(x) - 1).
(5) A'(x - Integral 2*A(x) dx) = (1 - A(x))/(1 - 2*A(x)).
a(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k.
Showing 1-6 of 6 results.

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