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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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.

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).

A280575 E.g.f. satisfies: A(x - Integral 4*A(x) dx) = x + Integral A(x) dx.

Original entry on oeis.org

1, 5, 85, 2625, 119225, 7209725, 547774525, 50342086425, 5448583486625, 680816992367125, 96720369562897125, 15433474653279056625, 2738894920523846767625, 536165521694896664958125, 114986014404299081313978125, 26857525461337375682549015625, 6797751676147997291181732240625, 1856223312322488721077123869053125, 544728880837527263669006279810003125, 171209466763535665605709125529165390625
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2017

Keywords

Examples

			E.g.f.: A(x) = x + 5*x^2/2! + 85*x^3/3! + 2625*x^4/4! + 119225*x^5/5! + 7209725*x^6/6! + 547774525*x^7/7! + 50342086425*x^8/8! + 5448583486625*x^9/9! + 680816992367125*x^10/10! + 96720369562897125*x^11/11! + 15433474653279056625*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - 4*G(x)) = x + G(x) where
G(x) = x^2/2! + 5*x^3/3! + 85*x^4/4! + 2625*x^5/5! + 119225*x^6/6! + 7209725*x^7/7! + 547774525*x^8/8! + 50342086425*x^9/9! + 5448583486625*x^10/10! + 680816992367125*x^11/11! + 96720369562897125*x^12/12! +...
Also, A(x) = x + 5 * G( (4*A(x) + x)/5 ).
RELATED SERIES.
We have (4*A(x) + x)/5 = Series_Reversion( x - Integral 4*A(x) dx ), where
(4*A(x) + x)/5 = x + 4*x^2/2! + 68*x^3/3! + 2100*x^4/4! + 95380*x^5/5! + 5767780*x^6/6! + 438219620*x^7/7! + 40273669140*x^8/8! + 4358866789300*x^9/9! + 544653593893700*x^10/10! + 77376295650317700*x^11/11! + 12346779722623245300*x^12/12! +...
Further, A( (4*A(x) + x)/5 ) = (A'(x) - 1)/(4*A'(x) + 1), which begins
A( (4*A(x) + x)/5 ) = x + 9*x^2/2! + 213*x^3/3! + 8365*x^4/4! + 463905*x^5/5! + 33459905*x^6/6! + 2985550445*x^7/7! + 318647659765*x^8/8! + 39707695580825*x^9/9! + 5672802966674825*x^10/10! + 916011286267596325*x^11/11! + 165283895506659660925*x^12/12! +...

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280571, A280572, A280573, A280574.

Programs

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