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 1-6 of 6 results.

A361307 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^3*A'(x)^4).

Original entry on oeis.org

1, 1, 15, 462, 20719, 1187628, 81575478, 6470236914, 578865763791, 57491440616067, 6266161502595672, 743009082083639748, 95191896469891628934, 13103364445591714775407, 1928820020328686200102278, 302383969785427961077318020, 50307405653295945234562827135
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^3 + 15*x^5 + 462*x^7 + 20719*x^9 + 1187628*x^11 + 81575478*x^13 + 6470236914*x^15 + 578865763791*x^17 + ... + a(n)*x^(2*n-1) + ...
By definition, A(x - x^3*A'(x)^4) = x, where
A'(x) = 1 + 3*x^2 + 75*x^4 + 3234*x^6 + 186471*x^8 + 13063908*x^10 + 1060481214*x^12 + 97053553710*x^14 + ... + A361537(n)*x^(2*n) + ...
Also,
A'(x) = 1 + (d/dx x^3*A'(x)^4) + (d^2/dx^2 x^6*A'(x)^8)/2! + (d^3/dx^3 x^9*A'(x)^12)/3! + (d^4/dx^4 x^12*A'(x)^16)/4! + (d^5/dx^5 x^15*A'(x)^20)/5! + ... + (d^n/dx^n x^(3*n)*A'(x)^(4*n))/n! + ...
Further,
A(x) = x * exp( x^2*A'(x)^4 + (d/dx x^5*A'(x)^8)/2! + (d^2/dx^2 x^8*A'(x)^12)/3! + (d^3/dx^3 x^11*A'(x)^16)/4! + (d^4/dx^4 x^14*A'(x)^20)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A'(x)^(4*n))/n! + ... ).

Links

Crossrefs

Cf. A361537.
Cf. A229619, A360976, A360977, A360978, A361302, A361308, A361309, A361310, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^3*A'^4 +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)^4).
(2) A(x) = x + A(x)^3 * A'(A(x))^4.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A'(x)^(4*n) / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A'(x)^(4*n) / n! is the g.f. of A361537.
(5) a(n) = A361537(n-1)/(2*n-1) for n >= 1.

A361308 G.f. A(x) satisfies A(x) = Series_Reversion(x - x^4*A'(x)).

Original entry on oeis.org

1, 1, 8, 122, 2676, 75197, 2548336, 100461956, 4500071172, 225305924896, 12456434569184, 753380353835754, 49473301917640864, 3505613955205438686, 266627715169575108168, 21667902182055638829520, 1873978995774161192935320, 171874439346918445003163152
Offset: 1

Views

Author

Paul D. Hanna, Mar 17 2023

Keywords

Examples

			G.f.: A(x) = x + x^4 + 8*x^7 + 122*x^10 + 2676*x^13 + 75197*x^16 + 2548336*x^19 + 100461956*x^22 + 4500071172*x^25 + ... + a(n)*x^(3*n-2) + ...
By definition, A(x - x^4*A'(x)) = x, where
A'(x) = 1 + 4*x^3 + 56*x^6 + 1220*x^9 + 34788*x^12 + 1203152*x^15 + 48418384*x^18 + 2210163032*x^21 + ... + A361541(n)*x^(3*n) + ...
Also,
A'(x) = 1 + (d/dx x^4*A'(x)) + (d^2/dx^2 x^8*A'(x)^2)/2! + (d^3/dx^3 x^12*A'(x)^3)/3! + (d^4/dx^4 x^16*A'(x)^4)/4! + (d^5/dx^5 x^20*A'(x)^5)/5! + ... + (d^n/dx^n x^(4*n)*A'(x)^n)/n! + ...
Further,
A(x) = x * exp( x^3*A'(x) + (d/dx x^7*A'(x)^2)/2! + (d^2/dx^2 x^11*A'(x)^3)/3! + (d^3/dx^3 x^15*A'(x)^4)/4! + (d^4/dx^4 x^19*A'(x)^5)/5! + ... + (d^(n-1)/dx^(n-1) x^(4*n-1)*A'(x)^n)/n! + ... ).

Links

Crossrefs

Cf. A361541.
Cf. A229619, A360976, A360977, A360978, A361302, A361307, A361309, A361310, A361311.

Programs

  • PARI
    {a(n) = my(A=x+x^3); for(i=1, n, A = serreverse(x - x^4*A' +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) A(x) = x + A(x)^4 * A'(A(x)).
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(4*n-1) * A'(x)^n / n! ).
(4) A'(x) = Sum_{n>=0} d^n/dx^n x^(4*n) * A'(x)^n / n! is the g.f. of A361541.
(5) a(n) = A361541(n-1)/(3*n-2) for n >= 1.

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.

A361536 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(3*n) / n!.

Original entry on oeis.org

1, 3, 60, 2037, 92187, 5066952, 322801089, 23197971285, 1848188250810, 161297106209607, 15285968218925460, 1562519987561305566, 171348519312001997550, 20068058089211306151393, 2500498134501774994768119, 330350627790472265384885061, 46136067767500181432129130897
Offset: 0

Views

Author

Paul D. Hanna, Mar 15 2023

Keywords

Examples

			G.f.: A(x) = 1 + 3*x^2 + 60*x^4 + 2037*x^6 + 92187*x^8 + 5066952*x^10 + 322801089*x^12 + 23197971285*x^14 + ... + a(n)*x^(2*n) + ...
where
A(x) = 1 + (d/dx x^3*A(x)^3) + (d^2/dx^2 x^6*A(x)^6)/2! + (d^3/dx^3 x^9*A(x)^9)/3! + (d^4/dx^4 x^12*A(x)^12)/4! + (d^5/dx^5 x^15*A(x)^15)/5! + ... + (d^n/dx^n x^(3*n)*A(x)^(3*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)^3), which begins
B(x) = x + x^3 + 12*x^5 + 291*x^7 + 10243*x^9 + 460632*x^11 + 24830853*x^13 + ... + A361302(n+1)*x^(2*n+1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^2*A(x)^3 + (d/dx x^5*A(x)^6)/2! + (d^2/dx^2 x^8*A(x)^9)/3! + (d^3/dx^3 x^11*A(x)^12)/4! + (d^4/dx^4 x^14*A(x)^15)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^(3*n))/n! + ... ).

Links

Crossrefs

Cf. A360950, A360974, A360975, A360973, A361046, A361537, A361541, A361542, A361543, A361551.
Cf. A361302.

Programs

  • Mathematica
    nmax = 20; r = 3; s = 3; A[] = 0; Do[A[x] = D[Normal[InverseSeries[x - x^r*A[x]^s + O[x]^k]], x], {k, 1, (r-1)*(nmax+1)+r}]; Table[CoefficientList[A[x], x][[j]], {j, 1, (r-1)*(nmax+1), r-1}] (* Vaclav Kotesovec, Mar 16 2023 *)
  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^(3*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^3 +O(x^(2*n+2))))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^3).
(3) B(x - x^3*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(3*n) / n! ) is the g.f. of A361302.
(4) a(n) = (2*n+1) * A361302(n+1) for n >= 0.
a(n) ~ c * 2^n * n! * n^((21*LambertW(1/3) - 1 + 3/(1 + LambertW(1/3)))/4) / LambertW(1/3)^n, where c = 0.123530460429388663183565497... - Vaclav Kotesovec, Mar 16 2023
Showing 1-6 of 6 results.

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