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 1-4 of 4 results.

A213591 G.f. A(x) satisfies A( x - A(x)^2 ) = x.

Original entry on oeis.org

1, 1, 4, 24, 178, 1512, 14152, 142705, 1528212, 17211564, 202460400, 2474708496, 31310415376, 408815254832, 5495451727376, 75907303147652, 1075685334980240, 15618612118252960, 232102241507321384, 3526880759915999016, 54755450619399484512, 867928449982022915984
Offset: 1

Views

Author

Paul D. Hanna, Jun 14 2012

Keywords

Comments

Unsigned version of A139702.
Self-convolution is A276370.
Row sums of triangle A277295.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
where A(x) = x + A(A(x))^2:
A(A(x)) = x + 2*x^2 + 10*x^3 + 69*x^4 + 568*x^5 + 5250*x^6 + 52792*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
The g.f. satisfies the series:
A(x) = x + A(x)^2 + d/dx A(x)^4/2! + d^2/dx^2 A(x)^6/3! + d^3/dx^3 A(x)^8/4! +...
Logarithmic series:
log(A(x)/x) = A(x)^2/x + [d/dx A(x)^4/x]/2! + [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! +...
Also, A(x) = x*G(A(x)^2/x) where G(x) = x/A(x/G(x)^2) is the g.f. of A212411:
G(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 + 15261*x^7 +...
Also, A(x)^2 = x*F(A(x)) where F(x) is the g.f. of A213628:
F(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 + 46013*x^8 +...

Links

Crossrefs

Cf. A220379, A139702, A212411, A138740, A213628, A213639.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.
Cf. A275765, A276360, A276361, A276362, A276363, A276370.
Cf. A277295 (triangle).

Programs

  • Mathematica
    terms = 22; A[] = 0; Do[A[x] = x + A[A[x]]^2 + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 09 2018 *)
  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^2+x*O(x^n))); polcoeff(A, n))}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1,21,print1(a(n),", "))
  • PARI
    b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 2*j)));
a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025

Formula

G.f. satisfies:
(1) A(x) = x + A(A(x))^2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(2*n)/x / n! ).
(4) A(x) = x*G(A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
(5) A(x)^2 = x*F(A(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628.
(6) x = A(A( x-x^2 - A(x)^2 )). - Paul D. Hanna, Jul 01 2012
(7) A(x) is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = x + B^2;
B = A + C^2;
C = B + D^2;
D = C + E^2; ...
where B = A(A(x)), C = A(A(A(x))), D = A(A(A(A(x)))), etc.
...
a(n) = Sum_{k=0..n-1} A277295(n,k).
From Seiichi Manyama, Jun 05 2025: (Start)
Let b(n,k) = [x^n] (A(x)/x)^k.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,2*j).
a(n) = b(n-1,1). (End)

A138740 G.f. satisfies A(x) = A(A(x)) - x^2 with A(0)=0.

Original entry on oeis.org

1, 1, -2, 9, -56, 420, -3572, 33328, -334354, 3559310, -39838760, 465743720, -5658983108, 71191948512, -924554859776, 12365546196641, -169995491295312, 2398380272232272, -34680290150700800, 513390937937217088, -7773229533145403728
Offset: 1

Views

Author

Paul D. Hanna, Mar 26 2008, Mar 27 2008, Apr 30 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-..;
A(A(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
The g.f. satisfies:
A(x) = x + x^2*exp((x-A(x))/x + [d/dx (x-A(x))^2/x]/2! + [d^2/dx^2 (x-A(x))^3/x]/3! + [d^3/dx^3 (x-A(x))^4/x]/4! +...)^2.
Higher order iterations of A=A(x) may be expressed in terms of A and x:
A(A(x)) = A + x^2 ;
A(A(A(x))) = (A + A^2) + x^2 ;
A(A(A(A(x)))) = (A + 2*A^2) + (1 + 2*A)*x^2 + x^4 ;
A(A(A(A(A(x))))) = (A + 3*A^2 + 2*A^3 + A^4) + (1 + 4*A + 2*A^2)*x^2 + 2*x^4 ;
A(A(A(A(A(A(x)))))) = (A + 4*A^2 + 6*A^3 + 5*A^4) + (1 + 6*A + 10*A^2 + 8*A^3)*x^2 + (3 + 6*A + 8*A^2)*x^4 + (2 + 4*A)*x^6 + x^8 ;
A(A(A(A(A(A(A(x))))))) = (A + 5*A^2 + 12*A^3 + 18*A^4 + 14*A^5 + 10*A^6 + 4*A^7 + A^8) + (1 + 8*A + 24*A^2 + 40*A^3 + 30*A^4 + 16*A^5 + 4*A^6)*x^2 + (4 + 18*A + 40*A^2 + 24*A^3 + 8*A^4)*x^4 + (6 + 20*A + 8*A^2)*x^6 + 5*x^8 .
The sums of coefficients in the above expansions form A000278: [1,1,2,3,7,16,65,321,4546,107587,20773703,...].
Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then
G(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
G(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
so that G(x)^2 = A(x) - x and G(x + G(x)^2) = x.

Crossrefs

Cf. A000278, A138739.
Cf. A139702, A276370.

Programs

  • PARI
    {a(n)=local(A=x+x^2);if(n<1,0, for(i=3,n,A=A-polcoeff(subst(A,x,A+x*O(x^i))-x^2,i)*x^i);polcoeff(A,n))}
  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1,n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(serreverse(A), n))}
  • PARI
    /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Mar 24 2011] */
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+x^2*exp(2*sum(m=0, n, Dx(m, (x-A)^(m+1)/x)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = x + x^2*exp( 2*Sum_{n>=0} [d^n/dx^n (x-A(x))^(n+1)/x]/(n+1)! ). - Paul D. Hanna, Mar 24 2011
Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then G(x)^2 = A(x) - x so that G(x + G(x)^2) = x.

Extensions

Edited by Paul D. Hanna, May 16 2010

A380558 G.f. A(x) satisfies A(x - A(x)) = x^2/(1 - x^2).

Original entry on oeis.org

1, 2, 10, 62, 469, 4028, 37984, 385202, 4144798, 46882400, 553733875, 6795347708, 86314711993, 1131422763410, 15268625617174, 211726229534738, 3012057754693912, 43903115899714844, 654923002676505376, 9989373316478767304, 155663132037403882606, 2476418549848925209424, 40195761790035415573939
Offset: 2

Views

Author

Paul D. Hanna, Feb 13 2025

Keywords

Comments

Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.

Examples

			G.f.: A(x) = x^2 + 2*x^3 + 10*x^4 + 62*x^5 + 469*x^6 + 4028*x^7 + 37984*x^8 + 385202*x^9 + 4144798*x^10 + 46882400*x^11 + 553733875*x^12 + ...
where A(x - A(x)) = x^2/(1 - x^2).
Let B(x) = Series_Reversion(x - A(x)), where
B(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1645*x^6 + 15652*x^7 + 160186*x^8 + 1739032*x^9 + 19838179*x^10 + ... + A380678(n)*x^n + ...
then B(x) = x + A(B(x)).

Links

Crossrefs

Cf. A380678, A276370, A004760.

Programs

  • PARI
    /* Generates N terms of this sequence */
N = 40; A=x^2; for(m=1,N, A=truncate(A); B = serreverse(x - A +x*O(x^m)); A = B^2/(1-B^2) ); Vec(A)

Formula

G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.
(1) A(x - A(x)) = x^2/(1 - x^2).
(2) A(x) = B(x)^2/(1 - B(x)^2) where B(x) = x + A(B(x)) and B(x - A(x)) = x.
(3) A(x) = B(x)^2/(1 - B(x)^2) where B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n!.

A292809 G.f. A(x) satisfies: A( 2*x - A(x) ) = 2*x - A(x) + x^2.

Original entry on oeis.org

1, 1, 2, 9, 56, 420, 3572, 33328, 334354, 3559310, 39838760, 465743720, 5658983108, 71191948512, 924554859776, 12365546196641, 169995491295312, 2398380272232272, 34680290150700800, 513390937937217088, 7773229533145403728, 120277760289804227632, 1900583166564027019136, 30649888151334972466392, 504153517331248726221392, 8454018409655883681321232, 144451967918022160558965408, 2513925490162481746629200624, 44542176917098830784415314624
Offset: 1

Views

Author

Paul D. Hanna, Sep 24 2017

Keywords

Comments

Apart from signs, essentially the same as A138740.
Apparently a(n) = A276370(n) wherever defined. - R. J. Mathar, Sep 26 2017

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 9*x^4 + 56*x^5 + 420*x^6 + 3572*x^7 + 33328*x^8 + 334354*x^9 + 3559310*x^10 + 39838760*x^11 + 465743720*x^12 + 5658983108*x^13 + 71191948512*x^14 + 924554859776*x^15 + 12365546196641*x^16 +...
such that A( 2*x - A(x) ) = 2*x - A(x) + x^2.

Links

Crossrefs

Cf. A138740, A276370, A292810.

Programs

  • PARI
    {a(n) = my(A=x, V=[1, 1]); for(i=1, n, V = concat(V, 0); A=x*Ser(V); V[#V] = Vec( subst(G=A, x, 2*x - A) )[#V]/(-1) ); V[n]}
for(n=1,30,print1(a(n),", "))
Showing 1-4 of 4 results.

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