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 1-5 of 5 results.

A171991 G.f. satisfies: A(x) = x + A(A(x))^2 + A(A(x))^4.

Original entry on oeis.org

1, 1, 4, 25, 190, 1644, 15636, 159977, 1736392, 19804646, 235758596, 2914539808, 37275052828, 491727695628, 6675266957376, 93073877729749, 1330851640325930, 19490214361291636, 292025342161401292, 4472427699080578554, 69958787931298226280, 1116911386287712897260
Offset: 1

Views

Author

Paul D. Hanna, Jun 27 2012

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1644*x^6 + 15636*x^7 +...
Related series begin:
A(A(x)) = x + 2*x^2 + 10*x^3 + 71*x^4 + 598*x^5 + 5634*x^6 + 57624*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 182*x^5 + 1580*x^6 + 15080*x^7 +...
A(A(x))^4 = x^4 + 8*x^5 + 64*x^6 + 556*x^7 + 5192*x^8 + 51536*x^9 +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 58*x^5 + 446*x^6 + 3868*x^7 + 36705*x^8 +...
A(x)^4 = x^4 + 4*x^5 + 22*x^6 + 152*x^7 + 1205*x^8 + 10564*x^9 +...
where the series reversion of the g.f. A(x) begins:
x - A(x)^2 - A(x)^4 = x - x^2 - 2*x^3 - 10*x^4 - 62*x^5 - 468*x^6 - 4020*x^7 -...

Crossrefs

Cf. A190761, A171992, A213591.

Programs

  • Mathematica
    terms = 23; A[] = 0; Do[A[x] = x + A[A[x]]^2 + A[A[x]]^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, May 04 2025 *)
  • PARI
    {a(n)=local(A=x+x^2,B=x); for(i=1, n, B=subst(A,x,A+x*O(x^n));A=x+B^2+B^4); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2-A^4+x*O(x^n))); polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. satisfies: A( x - A(x)^2 - A(x)^4 ) = x.

A211794 G.f. satisfies: A(x) = x + A(A(x))^2 + A(A(x))^3.

Original entry on oeis.org

1, 1, 5, 34, 284, 2698, 28116, 314558, 3726504, 46310523, 599691087, 8051875596, 111674902936, 1595269110991, 23416737953975, 352551682694096, 5435771974479840, 85722307390977058, 1381216396128765272, 22718307467700650259, 381156536404199989205
Offset: 1

Views

Author

Paul D. Hanna, Jun 27 2012

Keywords

Examples

			G.f.: A(x) = x + x^2 + 5*x^3 + 34*x^4 + 284*x^5 + 2698*x^6 + 28116*x^7 + ...
Related series begin:
A(A(x)) = x + 2*x^2 + 12*x^3 + 94*x^4 + 872*x^5 + 9026*x^6 + 101194*x^7 + ...
A(A(x))^2 = x^2 + 4*x^3 + 28*x^4 + 236*x^5 + 2264*x^6 + 23796*x^7 + ...
A(A(x))^3 = x^3 + 6*x^4 + 48*x^5 + 434*x^6 + 4320*x^7 + 46302*x^8 + ...
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 78*x^5 + 661*x^6 + 6304*x^7 + 65624*x^8 + ...
A(x)^3 = x^3 + 3*x^4 + 18*x^5 + 133*x^6 + 1146*x^7 + 10995*x^8 + ...
where the series reversion R(x) of the g.f. A(x) begins:
R(x) = x - A(x)^2 - A(x)^3 = x - x^2 - 3*x^3 - 14*x^4 - 96*x^5 - 794*x^6 - 7450*x^7 - 76619*x^8 - 846161*x^9 - 9901282*x^10 + ...
Also, the series reversion of A(A(x)) is given by
x - x^2 - x^3 - A(x)^2 - A(x)^3 = x - 2*x^2 - 4*x^3 - 14*x^4 - 96*x^5 - 794*x^6 - 7450*x^7 - 76619*x^8 - 846161*x^9 - 9901282*x^10 - ...
Further, the series reversion of A(A(A(x))) starts as
x - x^2 - x^3 - A(x)^2 - A(x)^3 - R(x)^2 - R(x)^3 = x - 3*x^2 - 3*x^3 - 6*x^4 - 68*x^5 - 614*x^6 - 5952*x^7 - 62456*x^8 - 699438*x^9 - 8270469*x^10 + ...

Links

Crossrefs

Cf. A190761, A171991, A213591.

Programs

  • PARI
    {a(n)=local(A=x+x^2, B=x); for(i=1, n, B=subst(A, x, A+x*O(x^n)); A=x+B^2+B^3); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2-A^3+x*O(x^n))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A( x - A(x)^2 - A(x)^3 ) = x.
(2) A( A( x - x^2 - x^3 - A(x)^2 - A(x)^3 ) ) = x. - Paul D. Hanna, Sep 07 2020
(3) A( A( A( x - x^2 - x^3 - A(x)^2 - A(x)^3 - R(x)^2 - R(x)^3 ) ) ) = x, where R(x) = x - A(x)^2 - A(x)^3 satisfies R(A(x)) = x. - Paul D. Hanna, May 06 2024
From Paul D. Hanna, Aug 07 2024: (Start)
(4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) * (1 + A(x))^n / n!.
(5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)*(1 + A(x))^n/x / n! ). (End)

A368562 Expansion of g.f. A(x) satisfying A(x - A(x)) = x^2 - x^3.

Original entry on oeis.org

1, 1, 4, 22, 144, 1064, 8623, 75267, 698898, 6843478, 70209485, 751028445, 8344927123, 96028777007, 1141700185040, 13996586957076, 176645358631455, 2291885063335367, 30533724487796062, 417268607544901628, 5843943369536347505, 83810410526002091163, 1229907906811449747716
Offset: 2

Views

Author

Paul D. Hanna, Dec 30 2023

Keywords

Examples

			G.f.: A(x) = x^2 + x^3 + 4*x^4 + 22*x^5 + 144*x^6 + 1064*x^7 + 8623*x^8 + 75267*x^9 + 698898*x^10 + 6843478*x^11 + 70209485*x^12 + ...
where A(x - A(x)) = x^2 - x^3.
RELATED SERIES.
Let B(x) be the g.f. of A190761, then A(x) = B(x)^2 - B(x)^3 where
B(x) = x + x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 592*x^6 + 4670*x^7 + ...
Also,
A(B(x)) = x^2 + 3*x^3 + 14*x^4 + 84*x^5 + 592*x^6 + 4670*x^7 + ...

Links

Crossrefs

Cf. A190761.

Programs

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

Formula

G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.
(1) A(x - A(x)) = x^2 - x^3.
(2) A(x) = B(x)^2 - B(x)^3 where B(x) = x + A(B(x)) is the g.f. of A190761 and B(x - A(x)) = x.

A171992 G.f. satisfies: A(x) = x + A(A(x))^2 - A(A(x))^4.

Original entry on oeis.org

1, 1, 4, 23, 166, 1380, 12684, 125857, 1328084, 14754242, 171338020, 2069009164, 25877555908, 334197713580, 4445788022944, 60800921601639, 853479846713406, 12280659254071964, 180929894848439516, 2726751302240331150, 42001984460083899448, 660800371941797598828
Offset: 1

Views

Author

Paul D. Hanna, Jun 27 2012

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 23*x^4 + 166*x^5 + 1380*x^6 + 12684*x^7 +...
Related series begin:
A(A(x)) = x + 2*x^2 + 10*x^3 + 67*x^4 + 538*x^5 + 4866*x^6 + 48000*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 174*x^5 + 1444*x^6 + 13224*x^7 +...
A(A(x))^4 = x^4 + 8*x^5 + 64*x^6 + 540*x^7 + 4856*x^8 + 46352*x^9 +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 54*x^5 + 394*x^6 + 3276*x^7 + 29985*x^8 +...
A(x)^4 = x^4 + 4*x^5 + 22*x^6 + 144*x^7 + 1085*x^8 + 9100*x^9 +...
where the series reversion of the g.f. A(x) begins:
x - A(x)^2 + A(x)^4 = x - x^2 - 2*x^3 - 8*x^4 - 50*x^5 - 372*x^6 - 3132*x^7 -...

Crossrefs

Cf. A190761, A171991, A213591.

Programs

  • PARI
    {a(n)=local(A=x+x^2,B=x); for(i=1, n, B=subst(A,x,A+x*O(x^n));A=x+B^2-B^4); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2+A^4+x*O(x^n))); polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. satisfies: A( x - A(x)^2 + A(x)^4 ) = x.

A210591 G.f. satisfies: A(x) = x + (A(x) - A(x)^2) * A(A(x)) where A(0)=0.

Original entry on oeis.org

1, 1, 2, 6, 24, 116, 636, 3823, 24729, 170187, 1236536, 9431875, 75202833, 624669675, 5390981898, 48229922730, 446459295023, 4269397134669, 42117578374712, 428090022559608, 4478159536973989, 48163581285504612, 532096844251876645, 6033134642314812383
Offset: 1

Views

Author

Paul D. Hanna, Mar 23 2012

Keywords

Comments

Compare g.f. to a g.f. C(x) of the Catalan numbers (A000108):
C(x) = x + C(x)*C(C(x)) - C(x)*C(C(x))^2 where C(x) = (1-sqrt(1-4*x))/2.

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 116*x^6 + 636*x^7 + 3823*x^8 +...
Related expansions.
A(A(x)) = x + 2*x^2 + 6*x^3 + 23*x^4 + 106*x^5 + 562*x^6 + 3316*x^7 +...
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 64*x^6 + 304*x^7 + 1636*x^8 +...
A(x) - A(x)^2 = x + x^4 + 8*x^5 + 52*x^6 + 332*x^7 + 2187*x^8 +...
The series reversion of g.f. A(x) begins:
x - (x-x^2)*A(x) = x - x^2 - x^4 - 4*x^5 - 18*x^6 - 92*x^7 - 520*x^8 -...

Crossrefs

Cf. A190761.

Programs

  • PARI
    {a(n)=local(A=x+x^2);for(i=1,n,A=serreverse(x-(x-x^2)*A +x*O(x^n)));polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. satisfies: A( x - (x-x^2)*A(x) ) = x.
G.f. satisfies: A( C(x) - x*A(C(x)) ) = C(x) where C(x) = (1-sqrt(1-4*x))/2 is a g.f. of the Catalan numbers.
Showing 1-5 of 5 results.

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