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 1-7 of 7 results.

A302100 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (3*k + 1).

Original entry on oeis.org

1, 1, 5, 41, 449, 6081, 98133, 1846377, 39888353, 977117825, 26839621829, 818332799593, 27443807417569, 1004188344449473, 39809506543659477, 1699473112658002089, 77716022374143303489, 3789578550994707778305, 196255782523222432943109, 10756748528551996006448553, 622036345094017435642828161, 37846075344692579622469742529
Offset: 0

Views

Author

Paul D. Hanna, Apr 09 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 41*x^3 + 449*x^4 + 6081*x^5 + 98133*x^6 + 1846377*x^7 + 39888353*x^8 + 977117825*x^9 + 26839621829*x^10 + ...
such that
A(x) = 1 + x*A(x) + 4*x^2*A(x)^2 + 28*x^3*A(x)^3 + 280*x^4*A(x)^4 + 3640*x^5*A(x)^5 + 58240*x^6*A(x)^6 + ... + x^n * A(x)^n * Product_{k=0..n-1} (3*k + 1) + ...

Links

Crossrefs

Cf. A007559, A088368, A301363, A302535, A302565.

Programs

  • PARI
    /* Series Reversion of Triple Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m * prod(k=0,m-1,3*k + 1)) +x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 4*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(3*(n-k+1)/2)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (3*k + 1).
(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A007559(n)*x^n, the o.g.f. of the triple factorials.
(3) A(x) = 1 + x*A(x)^2 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)).
(4) A(x) = 1/(1 - x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - 6*x*A(x)/(1 - 7*x*A(x)/(1 - 9*x*A(x)/(1 - 10*x*A(x)/(1 - ...)))))))), a continued fraction.
a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 1/6) / (Gamma(1/3) * exp(n - 1/3)). - Vaclav Kotesovec, Jun 18 2019

A302535 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).

Original entry on oeis.org

1, 1, 6, 61, 846, 14746, 310016, 7665141, 218827766, 7106293246, 259169817316, 10497928495506, 467768758203676, 22739720141372196, 1197560448125948596, 67910602688355999461, 4125144974025630599846, 267199960610924528490486, 18382741943990196237909476, 1338585578875261292134492646, 102848696213697953204782043556
Offset: 0

Views

Author

Paul D. Hanna, Apr 09 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 61*x^3 + 846*x^4 + 14746*x^5 + 310016*x^6 + 7665141*x^7 + 218827766*x^8 + 7106293246*x^9 + 259169817316*x^10 + ...
such that
A(x) = 1 + x*A(x) + 5*x^2*A(x)^2 + 45*x^3*A(x)^3 + 585*x^4*A(x)^4 + 9945*x^5*A(x)^5 + 208845*x^6*A(x)^6 + ... + x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1) + ...

Links

Crossrefs

Cf. A007696, A088368, A301363, A302100, A302565.

Programs

  • PARI
    /* Series Reversion of Quartic Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m*prod(k=1,m-1,4*k + 1))+x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 5*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(4*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (4*k + 1).
(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A007696(n)*x^n, the o.g.f. of the quartic factorials.
(3) A(x) = 1 + x*A(x)^2 * (A(x) + 5*x*A'(x)) / (A(x) + x*A'(x)).
(4) A(x) = 1/(1 - x*A(x)/(1 - 4*x*A(x)/(1 - 5*x*A(x)/(1 - 8*x*A(x)/(1 - 9*x*A(x)/(1 - 12*x*A(x)/(1 - 13*x*A(x)/(1 - ...)))))))), a continued fraction.
a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n - 1/4)). - Vaclav Kotesovec, Jun 18 2019

A302565 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1).

Original entry on oeis.org

1, 1, 7, 85, 1429, 30517, 792007, 24293389, 862902745, 34918162057, 1587910815271, 80217252865861, 4457823231346717, 270261899977497325, 17749585402744292215, 1255201826997862952845, 95083758340337074058545, 7680863233559647281837265, 659040900304099125516970375, 59855299015030039092312638965
Offset: 0

Views

Author

Paul D. Hanna, Apr 09 2018

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 85*x^3 + 1429*x^4 + 30517*x^5 + 792007*x^6 + 24293389*x^7 + 862902745*x^8 + 34918162057*x^9 + ...
such that
A(x) = 1 + x*A(x) + 6*x^2*A(x)^2 + 66*x^3*A(x)^3 + 1056*x^4*A(x)^4 + 22176*x^5*A(x)^5 + ... + x^n*A(x)^n * Product_{k=0..n-1} (5*k + 1) + ...

Links

Crossrefs

Cf. A008548, A088368, A301363, A302100, A302535.

Programs

  • PARI
    /* Series Reversion of Quintuple Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m * prod(k=0, m-1, 5*k + 1)) +x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 6*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(5*floor(3*(n-k+1)/2)/3)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (5*k + 1).
(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A008548(n)*x^n, the o.g.f. of the quintuple factorials.
(3) A(x) = 1 + x*A(x)^2 * (A(x) + 6*x*A'(x)) / (A(x) + x*A'(x)).
(4) A(x) = 1/(1 - x*A(x)/(1 - 5*x*A(x)/(1 - 6*x*A(x)/(1 - 10*x*A(x)/(1 - 11*x*A(x)/(1 - 15*x*A(x)/(1 - 16*x*A(x)/(1 - ...)))))))), a continued fraction.
a(n) ~ sqrt(2*Pi) * 5^n * n^(n - 3/10) / (Gamma(1/5) * exp(n - 1/5)). - Vaclav Kotesovec, Jun 18 2019

A302102 G.f. A(x) satisfies: A(x) = 1 + x * (x*A(x)^4)' / (x*A(x))'.

Original entry on oeis.org

1, 1, 6, 60, 796, 12873, 243648, 5274630, 128693820, 3501032280, 105278025690, 3473075317080, 124851287129604, 4860913404292030, 203839755469977840, 9161170785398640570, 439317870410796482460, 22390147212903891054540, 1208526686930226056003640, 68866979112831866042953128, 4131396903316522288744806384, 260262349821990852529147365849
Offset: 0

Views

Author

Paul D. Hanna, Apr 07 2018

Keywords

Comments

Compare to: C(x) = 1 + x * (x*C(x)^2)' / (x*C(x))' holds when C(x) = 1 + x*C(x)^2, which is a g.f. of the Catalan numbers (A000108).
Compare to: G(x) = 1 + x * (x*G(x)^3)' / (x*G(x))' holds when G(x) = 1/(1 - x*G(x)/(1 - 2*x*G(x)/(1 - 3*x*G(x)/(1 - ...)))), a continued fraction, which is the g.f. of A301363.

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 60*x^3 + 796*x^4 + 12873*x^5 + 243648*x^6 + 5274630*x^7 + 128693820*x^8 + 3501032280*x^9 + 105278025690*x^10 + ...
such that A(x) = 1 + x * (x*A(x)^4)' / (x*A(x))'.

Crossrefs

Cf. A301363, A302101.

Programs

  • PARI
    /* Differential Equation */
{a(n) = my(A=1); for(i=0,n, A = 1 + x*(x*A^4)'/(x*A +x^2*O(x^n))'); polcoeff(A,n)}
for(n=0, 30, print1(a(n),", "))

Formula

G.f. A(x) satisfies: A(x) = 1 + x*A(x)^3 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)).

A301833 G.f. A(x) satisfies: A(x) = 1/(1 - 2*x*A(x)/(1 - 2*x*A(x)/(1 - 4*x*A(x)/(1 - 4*x*A(x)/(1 - 6*x*A(x)/(1 - 6*x*A(x)/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, 2, 12, 104, 1104, 13472, 183488, 2749056, 44996864, 802443776, 15579089920, 329170937856, 7562372632576, 188526816632832, 5083702487990272, 147676990509580288, 4600624321049722880, 153012055369679241216, 5409813656756850262016, 202534832564335070937088, 8001606648308588124045312
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 27 2018

Keywords

Examples

			G.f. A(x) = 1 + 2*x + 12*x^2 + 104*x^3 + 1104*x^4 + 13472*x^5 + 183488*x^6 + 2749056*x^7 + 44996864*x^8 + ...
log(A(x)) = 2*x + 20*x^2/2 + 248*x^3/3 + 3472*x^4/4 + 53152*x^5/5 + 878144*x^6/6 + ... + A293471(n)*x^n/n + ...

Links

Crossrefs

Cf. A000165, A088368, A293471, A301363.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + Sum[(2*k)!!*x^k, {k, 1, n}])^(n+1)/(n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2021 *)

Formula

a(n) = [x^n] (Sum_{k>=0} A000165(k)*x^k)^(n+1)/(n + 1).
a(n) ~ sqrt(Pi) * (2*n)^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Nov 05 2021

A301834 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x)/(1 - 4*x*A(x)/(1 - 9*x*A(x)/(1 - 16*x*A(x)/(1 - ... - k^2*x*A(x)/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 6, 77, 1710, 59882, 3091200, 222190789, 21227659638, 2599346122814, 396581942797668, 73721984076543090, 16398099489074850108, 4299479561194904805396, 1312142733349302902243508, 461104766297721671082897333, 184846637953491751729984324518, 83842823980101547405726058204534
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 27 2018

Keywords

Examples

			G.f. A(x) = 1 + x + 6*x^2 + 77*x^3 + 1710*x^4 + 59882*x^5 + 3091200*x^6 + 222190789*x^7 + 21227659638*x^8 + ...

Crossrefs

Cf. A000364, A301363.

Programs

  • Mathematica
    Table[SeriesCoefficient[(1 + Sum[Abs[EulerE[2*k]]*x^k, {k, 1, n}])^(n+1)/(n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 05 2021 *)

Formula

a(n) = [x^n] (Sum_{k>=0} A000364(k)*x^k)^(n+1)/(n + 1).
a(n) ~ 2^(4*n + 3) * n^(2*n + 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Nov 05 2021

A305537 G.f. A(x) satisfies: A(x) = 1/(1 - x*A(x) - x*A(x)/(1 - x*A(x) - 2*x*A(x)/(1 - x*A(x) - 3*x*A(x)/(1 - x*A(x) - 4*x*A(x)/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 2, 11, 87, 844, 9438, 118217, 1636078, 24869591, 414422424, 7568815758, 151468591827, 3317061005044, 79265498450882, 2058189152006115, 57777549430984983, 1744191365957251044, 56332730020388347302, 1937412176139535240463, 70659708678402399722656
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 04 2018

Keywords

Examples

			G.f. A(x) = 1 + 2*x + 11*x^2 + 87*x^3 + 844*x^4 + 9438*x^5 + 118217*x^6 + 1636078*x^7 + 24869591*x^8 + ...

Links

Crossrefs

Cf. A001515, A027307, A301363.

Formula

a(n) = [x^n] (Sum_{k>=0} A001515(k)*x^k)^(n+1)/(n + 1).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.