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 +...
		

Crossrefs

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)

A139702 G.f. satisfies: x = A( x + A(x)^2 ).

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
Offset: 1

Views

Author

Paul D. Hanna, Apr 30 2008, May 20 2008

Keywords

Comments

Signed version of A213591.

Examples

			G.f.: A(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
where A(x + A(x)^2) = x.
Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then:
G(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... and
G(G(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+...
so that G(x) = G(G(x)) - x^2 = g.f. of A138740.
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! -+...
		

Crossrefs

Programs

  • Mathematica
    nmax = 20; sol = {a[1] -> 1}; nmin = Length[sol]+1;
    Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[x - A[x + A[x]^2] + O[x]^(n+1), x][[nmin;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nmin, nmax}];
    a /@ Range[nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)
  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1,n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(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, Dec 18 2010] */
    {a(n)=local(A=x-x^2+x*O(x^n));for(i=1,n,
    A=x*exp(sum(m=0,n,(-1)^(m+1)*Dx(m,A^(2*m+2)/x)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}

Formula

Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then G(x) = G(G(x)) - x^2 = g.f. of A138740.
G.f. satisfies: 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.
G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
A = 1 - x*B^2;
B = A - x*C^2;
C = B - x*D^2;
D = C - x*E^2;
E = D - x*F^2; ...
G.f. satisfies: A(x) = x*exp( Sum_{n>=0} (-1)^(n+1)*[d^n/dx^n A(x)^(2n+2)/x]/(n+1)! ). [Paul D. Hanna, Dec 18 2010]

A276370 G.f. A(x) satisfies: A( x - A(x) ) = x^2.

Original entry on oeis.org

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
Offset: 2

Views

Author

Paul D. Hanna, Sep 09 2016

Keywords

Comments

An unsigned version of A138740 (apart from initial term).
Self-convolution of A213591.

Examples

			G.f.: A(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 +...
such that A( x - A(x) ) = x^2.
		

Crossrefs

Programs

  • PARI
    {a(n) = local(A=x^2); for(i=1, n, A = serreverse(x - A +x*O(x^n))^2); polcoeff(A, n)}
    for(n=2,30,print1(a(n),", "))
    
  • PARI
    {Dx(n, F) = local(D=F); for(i=1, n, D=deriv(D)); D}
    {a(n) = local(A=x^2 +x*O(x^n)); for(i=1, n, A = (x + sum(m=1, n, Dx(m-1, A^m)/m!) +x*O(x^n))^2); polcoeff(A, n)}
    for(n=2,30,print1(a(n),", "))

Formula

G.f. satisfies:
(1) A(x) = ( x + A( sqrt(A(x)) ) )^2.
(2) A(x) = ( x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n! )^2.
(3) A(x) = x^2 * exp( 2*Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n/x / n! ).
(4) A(x) = x^2 * G( A(x)/x )^2 where G(x) = 1 + x*G( 1 - 1/G(x) )^2 is the g.f. of A212411.
(5) A(x) = x^2 * F( sqrt(A(x)) )^2 where F(x) = 1 - (x^2/F(x)) / F( x^2/F(x) ) is the g.f. of A213628.

A213628 G.f. satisfies: A(x) = 1 - (x^2/A(x)) / A( x^2/A(x) ).

Original entry on oeis.org

1, 1, 3, 14, 85, 616, 5072, 46013, 450739, 4702265, 51731956, 595874703, 7147366614, 88905147730, 1143097097833, 15152617826426, 206646826047563, 2894398418226395, 41577147999077079, 611779190051375147, 9211548488261257610, 141802624561414800815
Offset: 1

Views

Author

Paul D. Hanna, Jun 16 2012

Keywords

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 +...
Related expansions:
x^2/A(x) = x - x^2 - 2*x^3 - 9*x^4 - 56*x^5 - 420*x^6 - 3572*x^7 -...
A(x^2/A(x)) = x - x^3 - 7*x^4 - 50*x^5 - 395*x^6 - 3436*x^7 -...
A(x) = x^2/Series_Reversion(G(x)) where G(x) is the g.f. of A213591:
G(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
such that G(x - G(x)^2) = x.
		

Crossrefs

Programs

  • PARI
    {a(n)=local(A=x,G=x); if(n<1, 0, for(i=1, n, G=serreverse(x - G^2+x*O(x^n)));A=x^2/(x-G^2);polcoeff(A, n))}
    for(n=1,25,print1(a(n),", "))

Formula

G.f.: A(x) = x^2/(x - G(x)^2) where G(x) is the g.f. of A213591 such that G(x^2/A(x)) = G(x - G(x)^2) = x.
G.f.: A(x) = Series_Reversion(x*F(x)) where F(x) = 1 + x*F(1 - 1/F(x))^2 is the g.f. of A212411.
Showing 1-4 of 4 results.