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.

Previous Showing 11-14 of 14 results.

A361309 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)^2).

Original entry on oeis.org

1, 1, 12, 294, 10556, 488105, 27237748, 1766404068, 129955274460, 10668008963012, 965419570076880, 95430263520948342, 10228351567332536636, 1181548204752647642190, 146354418172125510269224, 19353257235976807395819160, 2721549078621826864159594548
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^4 + 12*x^7 + 294*x^10 + 10556*x^13 + 488105*x^16 + 27237748*x^19 + 1766404068*x^22 + 129955274460*x^25 + ... + a(n)*x^(3*n-2) + ...
By definition, A(x - x^4*A'(x)^2) = x, where
A'(x) = 1 + 4*x^3 + 84*x^6 + 2940*x^9 + 137228*x^12 + 7809680*x^15 + 517517212*x^18 + 38860889496*x^21 + ... + A361542(n)*x^(3*n) + ...
Also,
A'(x) = 1 + (d/dx x^4*A'(x)^2) + (d^2/dx^2 x^8*A'(x)^4)/2! + (d^3/dx^3 x^12*A'(x)^6)/3! + (d^4/dx^4 x^16*A'(x)^8)/4! + (d^5/dx^5 x^20*A'(x)^10)/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^(2*n))/n! + ...
Further,
A(x) = x * exp( x^3*A'(x)^2 + (d/dx x^7*A'(x)^4)/2! + (d^2/dx^2 x^11*A'(x)^6)/3! + (d^3/dx^3 x^15*A'(x)^8)/4! + (d^4/dx^4 x^19*A'(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^(2*n))/n! + ... ).

Links

Crossrefs

Cf. A361542.
Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361310, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A'^2 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^4*A'(x)^2).
(2) A(x) = x + A(x)^4 * A'(A(x))^2.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^(2*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^(2*n) / n! is the g.f. of A361542.
(5) a(n) = A361542(n-1)/(3*n-2) for n >= 1.

A361310 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)^3).

Original entry on oeis.org

1, 1, 16, 538, 26676, 1705373, 131524408, 11778395196, 1195433981028, 135247561603456, 16853285080609312, 2292048750536003426, 337754031605269049112, 53608164572529006153454, 9118712400086550140230888, 1655104918901340697851158384, 319341008921919836189242604080
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^4 + 16*x^7 + 538*x^10 + 26676*x^13 + 1705373*x^16 + 131524408*x^19 + 11778395196*x^22 + ... + a(n)*x^(3*n-2) + ...
By definition, A(x - x^4*A'(x)^3) = x, where
A'(x) = 1 + 4*x^3 + 112*x^6 + 5380*x^9 + 346788*x^12 + 27285968*x^15 + 2498963752*x^18 + 259124694312*x^21 + ... + A361543(n)*x^(3*n) + ...
Also,
A'(x) = 1 + (d/dx x^4*A'(x)^3) + (d^2/dx^2 x^8*A'(x)^6)/2! + (d^3/dx^3 x^12*A'(x)^9)/3! + (d^4/dx^4 x^16*A'(x)^12)/4! + (d^5/dx^5 x^20*A'(x)^15)/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^(3*n))/n! + ...
Further,
A(x) = x * exp( x^3*A'(x)^3 + (d/dx x^7*A'(x)^6)/2! + (d^2/dx^2 x^11*A'(x)^9)/3! + (d^3/dx^3 x^15*A'(x)^12)/4! + (d^4/dx^4 x^19*A'(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^(3*n))/n! + ... ).

Links

Crossrefs

Cf. A361543.
Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361309, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A'^3 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^4*A'(x)^3).
(2) A(x) = x + A(x)^4 * A'(A(x))^3.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^(3*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^(3*n) / n! is the g.f. of A361543.
(5) a(n) = A361543(n-1)/(3*n-2) for n >= 1.

A361311 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^5*A'(x)).

Original entry on oeis.org

1, 1, 10, 195, 5520, 201255, 8881551, 457227585, 26805712005, 1759840463070, 127784731466660, 10164274303786460, 878859905526721250, 82080454974318915935, 8235485665033295289810, 883569144560890419421630, 100952601749463417250801935, 12239031817482031919864850550
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^5 + 10*x^9 + 195*x^13 + 5520*x^17 + 201255*x^21 + 8881551*x^25 + 457227585*x^29 + ... + a(n)*x^(4*n-3) + ...
By definition, A(x - x^5*A'(x)) = x, where
A'(x) = 1 + 5*x^4 + 90*x^8 + 2535*x^12 + 93840*x^16 + 4226355*x^20 + 222038775*x^24 + ... + A361551(n)*x^(4*n) + ...
Also,
A'(x) = 1 + (d/dx x^5*A'(x)) + (d^2/dx^2 x^10*A'(x)^2)/2! + (d^3/dx^3 x^15*A'(x)^3)/3! + (d^4/dx^4 x^20*A'(x)^4)/4! + (d^5/dx^5 x^25*A'(x)^5)/5! + ... + (d^n/dx^n x^(5*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^4*A'(x) + (d/dx x^9*A'(x)^2)/2! + (d^2/dx^2 x^14*A'(x)^3)/3! + (d^3/dx^3 x^19*A'(x)^4)/4! + (d^4/dx^4 x^24*A'(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(5*n-1)*A'(x)^n)/n! + ... ).

Links

Crossrefs

Cf. A361551.
Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361308, A361309, A361310.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^5*A' +x*O(x^(4*n)))); polcoeff(A, 4*n-3)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(4*n-3) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^5*A'(x)).
(2) A(x) = x + A(x)^5 * A'(A(x)).
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(5*n-1) * A'(x)^n / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(5*n) * A'(x)^n / n! is the g.f. of A361551.
(5) a(n) = A361551(n-1)/(4*n-3) for n >= 1.

A361047 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion(x - x^3*A'(x)^2).

Original entry on oeis.org

1, 1, 9, 159, 4051, 131688, 5132793, 231332589, 11778989157, 666865748751, 41494745678544, 2812781975630049, 206264308294757115, 16268935714201604701, 1373512281722006688063, 123601628009085259269819, 11812339040349301277253801, 1194940136210629914238593762
Offset: 1

Views

Author

Paul D. Hanna, Mar 03 2023

Keywords

Comments

Conjecture: a(n) == 1 (mod 3) iff n = (3^k - 1)/2 for k >= 0, otherwise a(n) == 0 (mod 3).

Examples

			G.f.: A(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^2) = x, where
A'(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + A361046(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^2) + (d^2/dx^2 x^6*A'(x)^4)/2! + (d^3/dx^3 x^9*A'(x)^6)/3! + (d^4/dx^4 x^12*A'(x)^8)/4! + (d^5/dx^5 x^15*A'(x)^10)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(2*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^2 + (d/dx x^5*A'(x)^4)/2! + (d^2/dx^2 x^8*A'(x)^6)/3! + (d^3/dx^3 x^11*A'(x)^8)/4! + (d^4/dx^4 x^14*A'(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(2*n))/n! + ... ).

Links

Crossrefs

Cf. A361046, A229619, A360977, A360978.

Programs

  • PARI
    {a(n) = my(A=x+x^2); for(i=1, n, A=serreverse(x - x^3*(A')^2 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1) may be defined by the following.
(1) A(x) = Series_Reversion(x - x^3*A'(x)^2).
(2) A(x) = x + A(x)^3 * A'(A(x))^2.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(2*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(2*n) / n!, where A'(x) is the g.f. of A361046.
a(n) = A361046(n-1)/(2*n-1) for n >= 1.
Previous Showing 11-14 of 14 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.