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 1-10 of 14 results. Next

A277411 Column 1 of triangle A277410.

Original entry on oeis.org

0, 3, 13, 38, 94, 213, 459, 960, 1972, 4007, 8089, 16266, 32634, 65385, 130903, 261956, 524080, 1048347, 2096901, 4194030, 8388310, 16776893, 33554083, 67108488, 134217324, 268435023, 536870449, 1073741330, 2147483122, 4294966737, 8589933999, 17179868556, 34359737704, 68719476035, 137438952733, 274877906166, 549755813070, 1099511626917, 2199023254651, 4398046510160
Offset: 1

Views

Author

Paul D. Hanna, Oct 25 2016

Keywords

Crossrefs

Cf. A277410, A277412.

Programs

  • PARI
    {A277410(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, 30, print1(A277410(n+1, 1), ", "));

Formula

Conjectures from Colin Barker, Nov 04 2016: (Start)
G.f.: x^2*(3-2*x) / ((1-x)^3*(1-2*x)).
a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4) for n>4.
a(n) = (8*(2^n-1)-n*(n+7))/2. (End)

A277412 A diagonal of triangle A277410.

Original entry on oeis.org

1, 13, 165, 2310, 36330, 640710, 12588345, 273544425, 6529047525, 170116046100, 4812116809500, 147071309685300, 4835838768886125, 170422360844360625, 6415409821472276625, 257182138353489599250, 10948868951071241940750, 493742086990731259931250, 23529007012831307040178125, 1182267810558397149214753125, 62507026744534189248771965625, 3470421725511913171914539625000, 201956614461150241288627906875000
Offset: 0

Views

Author

Paul D. Hanna, Oct 25 2016

Keywords

Comments

The adjacent diagonal in triangle A277410 forms the odd double factorials.

Crossrefs

Cf. A277410, A277411.

Programs

  • PARI
    {A277410(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=0, 30, print1(A277410(n+3, n), ", "));

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).

A279843 E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 2*A(x) dx.

Original entry on oeis.org

1, 3, 18, 189, 2907, 59373, 1520019, 46964934, 1705072680, 71304998301, 3382510434561, 179805942701262, 10604941134768027, 688310036217216666, 48823571643364894410, 3762575719966561217301, 313432935903428395412205, 28098727418570995251538128, 2700377607104440375587008499, 277246288187233901613660728700
Offset: 1

Views

Author

Paul D. Hanna, Dec 29 2016

Keywords

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 18*x^3/3! + 189*x^4/4! + 2907*x^5/5! + 59373*x^6/6! + 1520019*x^7/7! + 46964934*x^8/8! + 1705072680*x^9/9! + 71304998301*x^10/10! + 3382510434561*x^11/11! + 179805942701262*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 2*G(x) where
G(x) = x^2/2! + 3*x^3/3! + 18*x^4/4! + 189*x^5/5! + 2907*x^6/6! + 59373*x^7/7! + 1520019*x^8/8! + 46964934*x^9/9! + 1705072680*x^10/10! +...
Also, A(x) = x + 3 * G( (A(x) + 2*x)/3 ).
RELATED SERIES.
We have (A(x) + 2*x)/3 = Series_Reversion( x - Integral A(x) dx ), where
(A(x) + 2*x)/3 = x + x^2/2! + 6*x^3/3! + 63*x^4/4! + 969*x^5/5! + 19791*x^6/6! + 506673*x^7/7! + 15654978*x^8/8! + 568357560*x^9/9! + 23768332767*x^10/10! + 1127503478187*x^11/11! + 59935314233754*x^12/12! +...
Further, A( (A(x) + 2*x)/3 ) = (A'(x) - 1)/(A'(x) + 2), which begins
A( (A(x) + 2*x)/3 ) = x + 4*x^2/2! + 33*x^3/3! + 441*x^4/4! + 8241*x^5/5! + 199071*x^6/6! + 5922360*x^7/7! + 209986506*x^8/8! + 8665824933*x^9/9! + 408861881955*x^10/10! + 21747689650404*x^11/11! +...

Links

Crossrefs

Cf. A277410, A210949, A277403, A279844, 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=1,q=2) = 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,1,2), ", "))
  • PARI
    /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n,p=1,q=2) = 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,1,2), ", "))
  • PARI
    /* Informal code to generate the first N terms: */
{N=20; p=1;q=2; 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 2*A(x) dx.
(2) A(x) = x + 3 * G( (A(x) + 2*x)/3 ), where G(x) = Integral A(x) dx.
(3) A(x) = -2*x + 3 * Series_Reversion(x - Integral A(x) dx).
(4) A( (A(x) + 2*x)/3 ) = (A'(x) - 1)/(A'(x) + 2).
(5) A'(x - Integral A(x) dx) = (1 + 2*A(x))/(1 - A(x)).
a(n) = Sum_{k=0..n-1} A277410(n,k) * 3^(n-k-1).

A277403 E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral A(x) dx.

Original entry on oeis.org

1, 2, 10, 90, 1190, 20930, 462070, 12326790, 386855630, 14000898310, 575440398330, 26532920708070, 1358954912773010, 76682330257445570, 4734315243483414890, 317932511564758225170, 23106045191162625194230, 1809303767549542227341490, 152057767850058496005946030, 13668688227104664304597942910, 1310201986290043690952261887230, 133552478071366935949713096470670, 14440878313638992240490923468851610
Offset: 1

Views

Author

Paul D. Hanna, Oct 14 2016

Keywords

Comments

a(n) is divisible by 10 for n>2 (conjecture).

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 90*x^4/4! + 1190*x^5/5! + 20930*x^6/6! + 462070*x^7/7! + 12326790*x^8/8! + 386855630*x^9/9! + 14000898310*x^10/10! +...
such that
A(x - Integral A(x) dx) = x + x^2/2! + 2*x^3/3! + 10*x^4/4! + 90*x^5/5! + 1190*x^6/6! + 20930*x^7/7! + 462070*x^8/8! +...+ a(n)*x^(n+1)/(n+1)! +...
which equals x + Integral A(x) dx.
RELATED SERIES.
Let G(x) = Integral A(x) dx, then
G( (A(x) + x)/2 ) = x^2/2! + 5*x^3/3! + 45*x^4/4! + 595*x^5/5! + 10465*x^6/6! + 231035*x^7/7! + 6163395*x^8/8! +...+ a(n)/2*x^n/n! +...
so that A(x) = x + 2 * G( (A(x) + x)/2 ).
A( (A(x) + x)/2 ) = x + 3*x^2/2! + 21*x^3/3! + 241*x^4/4! + 3885*x^5/5! + 81185*x^6/6! + 2093735*x^7/7! + 64463245*x^8/8! + 2313446975*x^9/9! + 95044136915*x^10/10! +...
which equals (A'(x) - 1)/(A'(x) + 1).

Links

Crossrefs

Cf. A277410, A210949, A279843.

Programs

  • Mathematica
    m = 24; A[_] = 0;
Do[G[x_] = Integrate[A[x], x]; A[x_] = x + 2 G[(A[x] + x)/2] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m-1]! // Rest (* Jean-François Alcover, Oct 20 2019 *)
  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F = x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - intformal(F)) - intformal(F), #A) ); n!*A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

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

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).

A280573 E.g.f. satisfies: A(x - Integral 2*A(x) dx) = x + Integral 3*A(x) dx.

Original entry on oeis.org

1, 5, 55, 1075, 30825, 1174725, 56153575, 3241453075, 219981653625, 17205716877125, 1527315775776375, 152004555650445875, 16793815038459239625, 2042866310966722613125, 271723598687954810434375, 39287423162026628955721875, 6143464129092882413626065625, 1034396495380447234136660853125, 186805274512176503194633726284375, 36060209533917578045193572845421875
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2017

Keywords

Examples

			E.g.f.: A(x) = x + 5*x^2/2! + 55*x^3/3! + 1075*x^4/4! + 30825*x^5/5! + 1174725*x^6/6! + 56153575*x^7/7! + 3241453075*x^8/8! + 219981653625*x^9/9! + 17205716877125*x^10/10! + 1527315775776375*x^11/11! + 152004555650445875*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 2*G(x)) = x + 3*G(x) where
G(x) = x^2/2! + 5*x^3/3! + 55*x^4/4! + 1075*x^5/5! + 30825*x^6/6! + 1174725*x^7/7! + 56153575*x^8/8! + 3241453075*x^9/9! + 219981653625*x^10/10! + 17205716877125*x^11/11! + 1527315775776375*x^12/12! +...
Also, A(x) = x + 5 * G( (2*A(x) + 3*x)/5 ).
RELATED SERIES.
We have (2*A(x) + 3*x)/5 = Series_Reversion( x - Integral 2*A(x) dx ), where
(2*A(x) + 3*x)/5 =  x + 2*x^2/2! + 22*x^3/3! + 430*x^4/4! + 12330*x^5/5! + 469890*x^6/6! + 22461430*x^7/7! + 1296581230*x^8/8! + 87992661450*x^9 + 6882286750850*x^10 + 610926310310550*x^11 + 60801822260178350*x^12 +...
Further, A( (2*A(x) + 3*x)/5 ) = (A'(x) - 1)/(2*A'(x) + 3), which begins
A( (2*A(x) + 3*x)/5 ) = x + 7*x^2/2! + 107*x^3/3! + 2665*x^4/4! + 93005*x^5/5! + 4201015*x^6/6! + 233920155*x^7/7! + 15535390105*x^8/8! + 1201670102125*x^9/9! + 106329616511975*x^10/10! + 10612821894707675*x^11/11! + 1181462628283585225*x^12/12! +...

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280571, A280572, A280574, A280575.

Programs

  • Mathematica
    m = 21; A[_] = 0;
Do[A[x_] = -3x/2 + 5/2 InverseSeries[x-Integrate[2A[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=2, q=3) = 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, 3), ", "))
  • PARI
    /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=2, q=3) = 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, 3), ", "))
  • PARI
    /* Informal code to generate the first N terms: */
{N=20; p=2; q=3; 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 3*A(x) dx.
(2) A(x) = x + 5 * G( (2*A(x) + 3*x)/5 ), where G(x) = Integral A(x) dx.
(3) A(x) = -3*x/2 + 5/2 * Series_Reversion(x - Integral 2*A(x) dx).
(4) A( (2*A(x) + 3*x)/5 ) = (A'(x) - 1)/(2*A'(x) + 3).
(5) A'(x - Integral 2*A(x) dx) = (1 + 3*A(x))/(1 - 2*A(x)).
a(n) = Sum_{k=0..n-1} A277410(n,k) * 2^k * 5^(n-k-1).

A280574 E.g.f. satisfies: A(x - Integral 3*A(x) dx) = x + Integral 2*A(x) dx.

Original entry on oeis.org

1, 5, 70, 1775, 66175, 3283475, 204594175, 15411893450, 1366394303500, 139767921720875, 16243630181913625, 2118892887756520250, 307173379745256857875, 49084564051462443496250, 8586127214178418541668750, 1634509914502001105016284375, 336910750825106071274158853125, 74862327518834451026921878887500, 17862833297180486514281227128971875, 4561279298680105599840369905594562500
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2017

Keywords

Examples

			E.g.f.: A(x) = x + 5*x^2/2! + 70*x^3/3! + 1775*x^4/4! + 66175*x^5/5! + 3283475*x^6/6! + 204594175*x^7/7! + 15411893450*x^8/8! + 1366394303500*x^9/9! + 139767921720875*x^10/10! + 16243630181913625*x^11/11! + 2118892887756520250*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 3*G(x)) = x + 2*G(x) where
G(x) = x^2/2! + 5*x^3/3! + 70*x^4/4! + 1775*x^5/5! + 66175*x^6/6! + 3283475*x^7/7! + 204594175*x^8/8! + 15411893450*x^9/9! + 1366394303500*x^10/10! + 139767921720875*x^11/11! + 16243630181913625*x^12/12! +...
Also, A(x) = x + 5 * G( (3*A(x) + 2*x)/5 ).
RELATED SERIES.
We have (3*A(x) + 2*x)/5 = Series_Reversion( x - Integral 3*A(x) dx ), where
(3*A(x) + 2*x)/5 = x + 3*x^2/2! + 42*x^3/3! + 1065*x^4/4! + 39705*x^5/5! + 1970085*x^6/6! + 122756505*x^7/7! + 9247136070*x^8/8! + 819836582100*x^9/9! + 83860753032525*x^10/10! + 9746178109148175*x^11/11! + 1271335732653912150*x^12/12! +...
Further, A( (3*A(x) + 2*x)/5 ) = (A'(x) - 1)/(3*A'(x) + 2), which begins
A( (3*A(x) + 2*x)/5 ) = x + 8*x^2/2! + 157*x^3/3! + 5075*x^4/4! + 230905*x^5/5! + 13636085*x^6/6! + 994743280*x^7/7! + 86697077570*x^8/8! + 8813260716925*x^9/9! + 1026216275720525*x^10/10! + 134948279040712300*x^11/11! + 19814992125974741525*x^12/12! +...

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280571, A280572, A280573, A280575.

Programs

  • Mathematica
    m = 21; A[_] = 0;
Do[A[x_] = -2x/3 + 5/3 InverseSeries[x-Integrate[3A[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=2) = 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, 2), ", "))
  • PARI
    /* A(x - Integral p*A(x) dx) = x + Integral q*A(x) dx: */
{a(n, p=3, q=2) = 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, 2), ", "))
  • PARI
    /* Informal code to generate the first N terms: */
{N=20; p=3; q=2; 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 2*A(x) dx.
(2) A(x) = x + 5 * G( (3*A(x) + 2*x)/5 ), where G(x) = Integral A(x) dx.
(3) A(x) = -2*x/3 + 5/3 * Series_Reversion(x - Integral 3*A(x) dx).
(4) A( (3*A(x) + 2*x)/5 ) = (A'(x) - 1)/(3*A'(x) + 2).
(5) A'(x - Integral 3*A(x) dx) = (1 + 2*A(x))/(1 - 3*A(x)).
a(n) = Sum_{k=0..n-1} A277410(n,k) * 3^k * 5^(n-k-1).
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