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 1-9 of 9 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)

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! -+...

Links

Crossrefs

Cf. A138740, A212411, A213591.
Cf. A088714, A088717, A091713, A120971, A140094, A140095.
Cf. A143426, A087949, A143435, A182969.

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]

A138739 G.f. A(x) satisfies: A(A(x)) = 3*A(x) - 2*x - x^2 with A(0)=0.

Original entry on oeis.org

1, 1, 2, 11, 88, 888, 10572, 143214, 2159154, 35702442, 640873656, 12394383780, 256762580460, 5671209169168, 133041670286160, 3304034094162183, 86616702087692256, 2390831825522972392, 69323685702986714272, 2107073248164657741448, 67003070810599639419680, 2225053954972969636237280, 77034579373254666948386880, 2776183496539544726567249520
Offset: 1

Views

Author

Paul D. Hanna, Mar 27 2008

Keywords

Comments

All self-compositions of A(x) may be expressed as a finite sum involving powers of A(x) and x.

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 11*x^4 + 88*x^5 + 888*x^6 + 10572*x^7 + 143214*x^8 + 2159154*x^9 + 35702442*x^10 + 640873656*x^11 + 12394383780*x^12 + 256762580460*x^13 + 5671209169168*x^14 + 133041670286160*x^15 +...
A(A(x)) = x + 2*x^2 + 6*x^3 + 33*x^4 + 264*x^5 + 2664*x^6 + 31716*x^7 + 429642*x^8 + 6477462*x^9 + 107107326*x^10 + 1922620968*x^11 + 37183151340*x^12 + 770287741380*x^13 + 17013627507504*x^14 + 399125010858480*x^15 +...
so that A(A(x)) + 2*x + x^2 = 3*A(x).
Self-compositions of A=A(x) may be expressed in terms of A and x:
A(A(x)) = 3*A - 2*x - x^2 ;
A(A(A(x))) = (7*A - A^2) - 6*x - 3*x^2 ;
A(A(A(A(x)))) = (15*A - 12*A^2) + (-14 + 12*A)*x +
(-11 + 6*A)*x^2 - 4*x^3 - x^4 ;
A(A(A(A(A(x))))) = (31*A - 83*A^2 + 14*A^3 - A^4) +
(-12*A^2 + 120*A - 30)*x + (-6*A^2 + 60*A - 63)*x^2 - 48*x^3 - 12*x^4 .

Links

Crossrefs

Cf. A138740.

Programs

  • PARI
    {a(n)=local(A=x+x^2);if(n<1,0, for(i=3,n+1,A=A+polcoeff(subst(A,x,A+x*O(x^i)),i)*x^i);polcoeff(A,n))}
for(n=1,20,print1(a(n),", "))

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.

Links

Crossrefs

Cf. A138740, A213591, A212411, A213628.

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.

A186262 Expansion of 3F2( 2, 1/2, 3/2; 3, 4;16 x).

Original entry on oeis.org

1, 2, 9, 56, 420, 3564, 33033, 327184, 3413124, 37119160, 417733316, 4837527072, 57397642640, 695394516600, 8579210711625, 107541060458400, 1367139314643300, 17599273282621800, 229116465142280100, 3013124257920348000, 39991185556010816400
Offset: 0

Views

Author

Olivier Gérard, Feb 16 2011

Keywords

Comments

Combinatorial interpretation welcome.
The sequence (n+3)*a(n) is the diagonal of the symmetric table b(m,n) = C(m+n,m)*C(m+n+2,m)*(m+3)/C(m+3,3). This table seems to have integer coefficients. - F. Chapoton, Jun 13 2025

Links

Crossrefs

Close to A138740.

Programs

  • Mathematica
    CoefficientList[Series[HypergeometricPFQ[{2, 1/2, 3/2}, {3, 4}, 16*x], {x, 0, 20}], x]
Table[Binomial[2*n,n]*Binomial[2*n+2,n]/Binomial[n+3,3],{n,0,20}] (* Vaclav Kotesovec, Oct 28 2012 *)

Formula

G.f. is equivalent to (-1 + 2F1(-3/2,-1/2;2;16*x) - 6*x*2F1(-1/2,1/2;3;16*x) )/(4*x^2).
a(n) = C(2*n,n)*C(2*n+2,n)/C(n+3,3). - Vaclav Kotesovec, Oct 28 2012
D-finite with recurrence +n*(n+3)*(n+2)*a(n) -4*(2*n+1)*(2*n-1)*(n+1)*a(n-1)=0. - R. J. Mathar, Feb 08 2021

A191557 G.f. satisfies: A(A(x))^2 = x^2 + 4*A(x)^3.

Original entry on oeis.org

1, 1, 1, -1, -5, 6, 57, -68, -996, 1151, 23487, -26316, -703858, 769268, 25912425, -27791388, -1146924362, 1212941187, 60112150656, -62911402588, -3686975047595, 3828485422340, 262043300715095, -270475215554448, -21394371719691000
Offset: 1

Views

Author

Paul D. Hanna, Jun 06 2011

Keywords

Comments

Compare the g.f. to the following property of G(x) = x*sqrt(1+4*x):
G(G(x))^2 = x^2 + 4*x^3 + 4*G(x)^3.

Examples

			G.f.: A(x) = x + x^2 + x^3 - x^4 - 5*x^5 + 6*x^6 + 57*x^7 - 68*x^8 +...
Note that x^3 is the only odd power of x in A(x)^2:
A(x)^2 = x^2 + 2*x^3 + 3*x^4 - 11*x^6 + 117*x^8 - 2001*x^10 +...
Illustrate A(A(x))^2 = x^2 + 4*A(x)^3 by the expansions:
A(A(x))^2 = x^2 + 4*x^3 + 12*x^4 + 24*x^5 + 16*x^6 - 60*x^7 - 72*x^8 + 640*x^9 + 768*x^10 - 11160*x^11 - 12916*x^12 +...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 4*x^6 - 15*x^7 - 18*x^8 + 160*x^9 + 192*x^10 - 2790*x^11 - 3229*x^12 +...
G.f. of odd bisection B(x) = (A(x) - A(-x))/2 begins:
B(x) = x + x^3 - 5*x^5 + 57*x^7 - 996*x^9 + 23487*x^11 +...
where A(x) = B(x) + x^3/B(x).

Crossrefs

Cf. A107700, A191565, A138740.

Programs

  • PARI
    {a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=A-(subst(A,x,A)-x*sqrt(1+4*A^3/x^2))/2);polcoeff(A,n)}

Formula

G.f. satisfies: A(x)^2 = A(-x)^2 + 4*x^3.
G.f. satisfies: A(-A(-x)) = x.
G.f. satisfies: A(x) = B(x) + x^3/B(x) where B(x) = (A(x) - A(-x))/2.

A191565 G.f. satisfies: A(A(x))^2 = A(x)^2 + 4*x^3.

Original entry on oeis.org

1, 2, -14, 184, -3194, 65472, -1503924, 37593664, -1004163802, 28314667072, -835650200380, 25652840146624, -815280469973380, 26728163562423360, -901336722528156712, 31194183364269262848, -1105930698812430437626
Offset: 1

Views

Author

Paul D. Hanna, Jun 06 2011

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 - 14*x^3 + 184*x^4 - 3194*x^5 + 65472*x^6 +...
Illustrate A(A(x))^2 - A(x)^2 = 4*x^3 with the expansions:
A(x)^2 = x^2 + 4*x^3 - 24*x^4 + 312*x^5 - 5456*x^6 + 113016*x^7 +...
A(A(x))^2 = x^2 + 8*x^3 - 24*x^4 + 312*x^5 - 5456*x^6 + 113016*x^7 +...
A(A(x)) = x + 4*x^2 - 20*x^3 + 236*x^4 - 3872*x^5 + 76716*x^6 - 1723488*x^7 +...
Let R(x) be the series reversion of A(x):
R(x) = x - 2*x^2 + 22*x^3 - 364*x^4 + 7390*x^5 - 170556*x^6 +...
R(x)^3 = x^3 - 6*x^4 + 78*x^5 - 1364*x^6 + 28254*x^7 - 655668*x^8 +...
where A(x)^2 = x^2 + 4*R(x)^3.

Crossrefs

Cf. A191557, 107700, A138740.

Programs

  • PARI
    {a(n)=local(A=x+x^2+x*O(x^n));for(i=1,n,A=A-(subst(A,x,A)-x*sqrt(4*x+A^2/x^2)));polcoeff(A,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),", "))

A139715 G.f. A(x) satisfies: A(x) = G(G(x)) where G(x) = x - A(x)^2 = g.f. of A139702.

Original entry on oeis.org

1, -2, 10, -69, 568, -5250, 52792, -566830, 6420640, -76095972, 938077528, -11975951312, 157808048792, -2140767942096, 29835756120952, -426490803168368, 6244476409802008, -93541594534237356, 1432261132629484052, -22397290780155132728
Offset: 1

Views

Author

Paul D. Hanna, Apr 30 2008

Keywords

Examples

			A(x) = x - 2*x^2 + 10*x^3 - 69*x^4 + 568*x^5 - 5250*x^6 + 52792*x^7 -+...
Let G(x) = x - A(x)^2 = g.f. of A139702:
G(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 + 14152*x^7 -+...
then A(x) = G(G(x)).

Crossrefs

Cf. A139702, A138740.

Programs

  • PARI
    {a(n)=local(A=x); if(n<1, 0, for(i=1,n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(subst(A,x,A+x*O(x^n)), n))}

Formula

Series_Reversion(A(x)) = F(F(x)) = F(x) + x^2 where F(x) = g.f. of A138740.
Showing 1-9 of 9 results.

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