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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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.

A280570 E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral 3*A(x) dx.

Original entry on oeis.org

1, 4, 28, 332, 5748, 131940, 3791692, 131375324, 5343640212, 250142552212, 13271217848604, 788346022938556, 51916178572447140, 3759254932421361284, 297243198474965188732, 25513664852425377663756, 2365246919693613357168916, 235776253411115081902083556, 25174157913006507920211300588, 2869108641038261410331666767772
Offset: 1

Views

Author

Paul D. Hanna, Jan 05 2017

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 28*x^3/3! + 332*x^4/4! + 5748*x^5/5! + 131940*x^6/6! + 3791692*x^7/7! + 131375324*x^8/8! + 5343640212*x^9/9! + 250142552212*x^10/10! + 13271217848604*x^11/11! + 788346022938556*x^12/12! +...
Let G(x) = Integral A(x) dx, then A(x - G(x)) = x + 3*G(x) where
G(x) = x^2/2! + 4*x^3/3! + 28*x^4/4! + 332*x^5/5! + 5748*x^6/6! + 131940*x^7/7! + 3791692*x^8/8! + 131375324*x^9/9! + 5343640212*x^10/10! + 250142552212*x^11/11! + 13271217848604*x^12/12! +...
Also, A(x) = x + 4 * G( (A(x) + 3*x)/4 ).
RELATED SERIES.
We have (A(x) + 3*x)/4 = Series_Reversion( x - Integral A(x) dx ), where
(A(x) + 3*x)/4 = x + x^2/2! + 7*x^3/3! + 83*x^4/4! + 1437*x^5/5! + 32985*x^6/6! + 947923*x^7/7! + 32843831*x^8/8! + 1335910053*x^9/9! + 62535638053*x^10/10! + 3317804462151*x^11/11! + 197086505734639*x^12/12! +...
Further, A( (A(x) + 3*x)/4 ) = (A'(x) - 1)/(A'(x) + 3), which begins
A( (A(x) + 3*x)/4 ) = x + 5*x^2/2! + 47*x^3/3! + 707*x^4/4! + 14825*x^5/5! + 401033*x^6/6! + 13340739*x^7/7! + 528281555*x^8/8! + 24323141773*x^9/9! + 1279128727141*x^10/10! + 75770789421947*x^11/11! + 4999463984999615*x^12 +...

Crossrefs

Cf. A277410, A210949, A277403, A279843, A279844, A279845, A280571, A280572, A280573, A280574, A280575.

Programs

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

A067146 Shifts left under exponential reversion.

Original entry on oeis.org

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

Views

Author

Christian G. Bower, Jan 03 2002

Keywords

Comments

A signed version of A210949.

Links

Crossrefs

Cf. A210949, A277410.

Formula

E.g.f. satisfies A^(-1)(x) = A'(x) - 1.
a(n) = (-1)^n * Sum_{k=0..n-1} A277410(n,k), for n>1. - Paul D. Hanna, Jan 06 2017
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