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.

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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)

A275765 G.f. satisfies: A(x - A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 2, 12, 106, 1148, 14156, 191400, 2775930, 42585412, 684496988, 11449962008, 198331811356, 3543990791480, 65136985937096, 1228531761076208, 23733123786608826, 468887742020767788, 9461919438245032500, 194817077269127033944, 4089069139317823277548, 87426000975842460304792, 1902787414323673070857528, 42133267254272433484761584, 948695717599714654940068604, 21712101305047777916075831096, 504865916349551192319293625016
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1148*x^5 + 14156*x^6 + 191400*x^7 + 2775930*x^8 + 42585412*x^9 + 684496988*x^10 + 11449962008*x^11 + 198331811356*x^12 +...
such that A(x - A(x)^2) = x + A(x)^2.
RELATED SERIES.
Series_Reversion(x - A(x)^2) = x + x^2 + 6*x^3 + 53*x^4 + 574*x^5 + 7078*x^6 + 95700*x^7 + 1387965*x^8 + 21292706*x^9 + 342248494*x^10 +...
which equals (A(x) + x)/2.
A( (A(x) + x)/2 ) = x + 3*x^2 + 22*x^3 + 221*x^4 + 2634*x^5 + 35086*x^6 + 506356*x^7 + 7773279*x^8 + 125441594*x^9 + 2110832382*x^10 +...
which equals sqrt( (A(x) - x)/2 ).
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 2*x^2 - 4*x^3 - 26*x^4 - 228*x^5 - 2396*x^6 - 28440*x^7 - 369114*x^8 - 5135468*x^9 - 75602108*x^10 - 1167066216*x^11 - 18768202924*x^12 +...
then Series_Reversion(x + A(x)^2) = x/2 + R(x)/2.

Links

Crossrefs

Cf. A277295, A213591, A276364, A276360, A276361, A276362, A276363.

Programs

  • Mathematica
    m = 26; A[_] = 0;
Do[A[x_] = x + 2 A[x/2 + A[x]/2]^2 + O[x]^(m+1) // Normal, {m+1}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1], F=x); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = -polcoeff(subst(F,x,x-F^2) - F^2,#A) );A[n]}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 2 * A( x/2 + A(x)/2 )^2.
(2) A(x) = -x + 2 * Series_Reversion(x - A(x)^2).
(3) R(x) = -x + 2 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/2 - R(x)/2 ) ) = x/2 + R(x)/2, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^(n-k).

A277295 G.f. A(x,y) satisfies: A( x - y*A(x,y)^2, y) = x + (1-y)*A(x,y)^2, where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 5, 14, 5, 0, 14, 74, 76, 14, 0, 42, 352, 698, 378, 42, 0, 132, 1588, 5088, 5404, 1808, 132, 0, 429, 6946, 32461, 56410, 37546, 8484, 429, 0, 1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0, 4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0, 16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0, 58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0
Offset: 1

Views

Author

Paul D. Hanna, Oct 11 2016

Keywords

Comments

More generally, we have the following related identity.
Given functions F and G with F(0)=0, F'(0)=1, G(0)=0, G'(0)=0,
if F(x - y*G(x)) = x + (1-y)*G(x), then
(1) F(x) = x + G( y*F(x) + (1-y)*x ),
(2) y*F(x) + (1-y)*x = Series_Reversion(x - y*G(x)),
(3) F(x) = x + G(x + y*G(x + y*G(x + y*G(x +...)))),
(4) F(x) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x)^n / n!.
The g.f. of this sequence A(x,y) equals F(x) in the above when G(x) = F(x)^2.

Examples

			G.f.: A(x,y)  = x + x^2 + (2*y + 2)*x^3 + (5*y^2 + 14*y + 5)*x^4 + (14*y^3 + 76*y^2 + 74*y + 14)*x^5 + (42*y^4 + 378*y^3 + 698*y^2 + 352*y + 42)*x^6 + (132*y^5 + 1808*y^4 + 5404*y^3 + 5088*y^2 + 1588*y + 132)*x^7 + (429*y^6 + 8484*y^5 + 37546*y^4 + 56410*y^3 + 32461*y^2 + 6946*y + 429)*x^8 + (1430*y^7 + 39446*y^6 + 244220*y^5 + 535410*y^4 + 486550*y^3 + 189940*y^2 + 29786*y + 1430)*x^9 + (4862*y^8 + 182732*y^7 + 1522466*y^6 + 4597402*y^5 + 6036632*y^4 + 3690410*y^3 + 1046190*y^2 + 126008*y + 4862)*x^10 +...
such that
A( x - y*A(x,y)^2, y)  =  x + (1-y)*A(x,y)^2.
Also,
A(x,y) = x + A( y*A(x,y) + (1-y)*x, y)^2.
...
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:
1;
1, 0;
2, 2, 0;
5, 14, 5, 0;
14, 74, 76, 14, 0;
42, 352, 698, 378, 42, 0;
132, 1588, 5088, 5404, 1808, 132, 0;
429, 6946, 32461, 56410, 37546, 8484, 429, 0;
1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0;
4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0;
16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0;
58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0; ...
RELATED SEQUENCES.
Given T(n,k) is the coefficient of x^n*y^k in g.f. A(x,y),
if b(n) = Sum_{k=0..n-1} T(n,k) * p^k * q^(n-k-1)
then B(x) = Sum_{n>=1} b(n)*x^n satisfies
(1) B(x - p*B(x)^2) = x + (q-p)*B(x)^2
(2) B(x)  =  x + B( p*B(x) + (q-p)*x )^2.
Examples:
A213591(n) = sum(k=0,n-1, T(n,k) )
A275765(n) = sum(k=0,n-1, T(n,k) * 2^(n-k) )
A276360(n) = sum(k=0,n-1, T(n,k) * 3^(n-k-1) )
A276361(n) = sum(k=0,n-1, T(n,k) * 2^k * 3^(n-k-1) )
A276362(n) = sum(k=0,n-1, T(n,k) * 4^(n-k-1) )
A276363(n) = sum(k=0,n-1, T(n,k) * 3^k * 4^(n-k-1) )
A276365(n) = sum(k=0,n-1, T(n,k) * 2^k )
A277300(n) = sum(k=0,n-1, T(n,k) * 5^(n-k-1) )
A277301(n) = sum(k=0,n-1, T(n,k) * 2^k * 5^(n-k-1) )
A277302(n) = sum(k=0,n-1, T(n,k) * 3^k * 5^(n-k-1) )
A277303(n) = sum(k=0,n-1, T(n,k) * 4^k * 5^(n-k-1) )
A277304(n) = sum(k=0,n-1, T(n,k) * 6^(n-k-1) )
A277305(n) = sum(k=0,n-1, T(n,k) * 5^k * 6^(n-k-1) )
A277306(n) = sum(k=0,n-1, T(n,k) * (-1)^k )
A277307(n) = sum(k=0,n-1, T(n,k) * 3^k )
A277308(n) = sum(k=0,n-1, T(n,k) * 3^k * 2^(n-k-1) )
A277309(n) = sum(k=0,n-1, T(n,k) * 5^k * 2^(n-k-1) )
A277310(n) = sum(k=0,n-1, T(n,k) * 4^k )
A277311(n) = sum(k=0,n-1, T(n,k) * 5^k )
...

Links

Crossrefs

Cf. A000108 (column 0), A138156 (column 1), A277296 (column 2), A277297 (diagonal), A277298 (central terms T(2*n-1,n-1)), A277299 (central terms T(2*n,n-1)).
Cf. A213591 (row sums), A275765, A276360, A276361, A276362, A276363, A276365.
Cf. A277300, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309, A277310, A277311.

Programs

  • Mathematica
    c[n_] := c[n] = Module[{A}, A[x_] = x; Do[A[x_] = x + A[y A[x] + (1-y) x + x O[x]^j]^2, {j, n}] // Normal; SeriesCoefficient[A[x], {x, 0, n}] // Expand];
T[n_, k_] := SeriesCoefficient[c[n], {y, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {T(n,k) = my(A=x); for(i=1, n, A = x + subst(A^2, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A,n,x),k,y)}
for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "));print(""))

Formula

G.f. A(x,y) also satisfies:
(1) A(x,y) = x + A( y*A(x,y) + (1-y)*x, y)^2.
(2) y*A(x,y) + (1-y)*x = Series_Reversion( x - y*A(x,y)^2 ).
(3) y*x + (1-y)*B(x,y) = Series_Reversion( x + (1-y)*A(x,y)^2 ), where B( A(x,y), y) = x.
(4) A(x,y) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) A(x,y)^(2*n) / n!.
In formulas 2 and 3, the series reversion is taken with respect to variable x.
T(n+1,0) = T(n+1,n-1) = binomial(2*n,n)/(n+1) = A000108(n) for n>=1.
T(n+1,1) = 4^n - (3*n+1)*binomial(2*n,n)/(n+1) = A138156(n-1) for n>=1.

A276360 G.f. satisfies: A(x - A(x)^2) = x + 2*A(x)^2.

Original entry on oeis.org

1, 3, 24, 276, 3858, 61092, 1056816, 19550475, 381543576, 7782820548, 164842646424, 3607654164924, 81281990795520, 1879865970374568, 44527769989124976, 1078220967132218616, 26650484274297181896, 671558570413109457264, 17234310756238557856200, 450044549619831325213920, 11949386806898017225833312, 322394088574898542428753168, 8833647058171126097908059720
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 24*x^3 + 276*x^4 + 3858*x^5 + 61092*x^6 + 1056816*x^7 + 19550475*x^8 + 381543576*x^9 + 7782820548*x^10 + 164842646424*x^11 + 3607654164924*x^12 +...
such that A(x - A(x)^2) = x + 2*A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - A(x)^2) = 2*x/3 + A(x)/3, which begins:
Series_Reversion(x - A(x)^2) = x + x^2 + 8*x^3 + 92*x^4 + 1286*x^5 + 20364*x^6 + 352272*x^7 + 6516825*x^8 + 127181192*x^9 + 2594273516*x^10 + 54947548808*x^11 +
1202551388308*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 3*x^2 - 6*x^3 - 51*x^4 - 564*x^5 - 7416*x^6 - 109764*x^7 - 1772028*x^8 - 30603930*x^9 - 558238326*x^10 - 10659285096*x^11 - 211688430204*x^12 +...
then Series_Reversion(x + 2*A(x)^2) = x/3 + 2*R(x)/3.

Links

Crossrefs

Cf. A277295, A213591, A275765, A276361, A276362, A276363.

Programs

  • Mathematica
    m = 24; A[_] = 0;
Do[A[x_] = x + 3 A[2 x/3 + A[x]/3]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^2) - 2*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 3 * A( 2*x/3 + A(x)/3 )^2.
(2) A(x) = -2*x + 3 * Series_Reversion(x - A(x)^2).
(3) 2*R(x) = -x + 3 * Series_Reversion(x + 2*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/3 - R(x)/3 ) ) = x/3 + 2*R(x)/3, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*3^(n-k-1).

A276361 G.f. satisfies: A(x - 2*A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 3, 30, 447, 8202, 171846, 3956796, 97916895, 2567551890, 70655670690, 2026596875268, 60282027684678, 1852444347792036, 58633762133405100, 1907098496516434680, 63620675921801106495, 2173457638433471757282, 75940916632597398212298, 2710857429948875567968692, 98775527832178103444182722, 3670845430153146908693608044, 139047871842184594320103381524, 5365224711989826990651317756232
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 30*x^3 + 447*x^4 + 8202*x^5 + 171846*x^6 + 3956796*x^7 + 97916895*x^8 + 2567551890*x^9 + 70655670690*x^10 + 2026596875268*x^11 + 60282027684678*x^12 +...
such that A(x - 2*A(x)^2) = x + A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 2*A(x)^2) = x/3 + 2*A(x)/3, which begins:
Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 20*x^3 + 298*x^4 + 5468*x^5 + 114564*x^6 + 2637864*x^7 + 65277930*x^8 + 1711701260*x^9 + 47103780460*x^10 + 1351064583512*x^11 + 40188018456452*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 3*x^2 - 12*x^3 - 132*x^4 - 1992*x^5 - 36144*x^6 - 742176*x^7 - 16688880*x^8 - 402824928*x^9 - 10300868160*x^10 - 276531035520*x^11 - 7742210941056*x^12 +...
then Series_Reversion(x + A(x)^2) = 2*x/3 + R(x)/3.

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276362, A276363.

Programs

  • Mathematica
    m = 24; A[_] = 0;
Do[A[x_] = x + 3 A[x/3 + 2 A[x]/3]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-2*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 3 * A( x/3 + 2*A(x)/3 )^2.
(2) 2*A(x) = -x + 3 * Series_Reversion(x - 2*A(x)^2).
(3) R(x) = -2*x + 3 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/3 - R(x)/3 ) ) = 2*x/3 + R(x)/3, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k*3^(n-k-1).

A276363 G.f. satisfies: A(x - 3*A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 4, 56, 1172, 30248, 892296, 28951344, 1010322900, 37384819496, 1452697058744, 58872642043856, 2475764515398568, 107619880380347920, 4821324372637921744, 222077355203506939104, 10497354682052048593332, 508414637258604924680136, 25197644191294099697736312, 1276547957544912412461457680, 66046883289153773427379134360, 3487101507192780951408327918192
Offset: 1

Views

Author

Paul D. Hanna, Aug 31 2016

Keywords

Examples

			G.f.: A(x) = x + 4*x^2 + 56*x^3 + 1172*x^4 + 30248*x^5 + 892296*x^6 + 28951344*x^7 + 1010322900*x^8 + 37384819496*x^9 + 1452697058744*x^10 + 58872642043856*x^11 +
2475764515398568*x^12 +...
such that A(x - 3*A(x)^2) = x + A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 3*A(x)^2) = x/4 + 3*A(x)/4, which begins:
Series_Reversion(x - 3*A(x)^2) = x + 3*x^2 + 42*x^3 + 879*x^4 + 22686*x^5 + 669222*x^6 + 21713508*x^7 + 757742175*x^8 + 28038614622*x^9 + 1089522794058*x^10 + 44154481532892*x^11 + 1856823386548926*x^12 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - 4*x^2 - 24*x^3 - 372*x^4 - 7944*x^5 - 204168*x^6 - 5942256*x^7 - 189500916*x^8 - 6490281480*x^9 - 235609789368*x^10 - 8983294304784*x^11 - 357373688297448*x^12 +...
then Series_Reversion(x + A(x)^2) = 3*x/4 + R(x)/4.

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362.

Programs

  • Mathematica
    m = 22; A[_] = 0;
Do[A[x_] = x + 4A[x/4 + 3A[x]/4]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 3*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 4 * A( x/4 + 3*A(x)/4 )^2.
(2) 3*A(x) = -x + 4 * Series_Reversion(x - 3*A(x)^2).
(3) R(x) = -3*x + 4 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/4 - R(x)/4 ) ) = 3*x/4 + R(x)/4, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k)*3^k*4^(n-k-1).

A277300 G.f. satisfies: A(x - A(x)^2) = x + 4*A(x)^2.

Original entry on oeis.org

1, 5, 60, 1000, 19970, 448160, 10926360, 283651245, 7740058300, 220046970860, 6476695275680, 196438030797880, 6117627849485360, 195082685133612800, 6355848358118392400, 211189970909192038500, 7146354688384980282000, 245970478274041025623200, 8602606263466490521359400, 305460999044315834902424200, 11003870605124169641012461600
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 60*x^3 + 1000*x^4 + 19970*x^5 + 448160*x^6 + 10926360*x^7 + 283651245*x^8 + 7740058300*x^9 + 220046970860*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • Mathematica
    m = 22; A[_] = 0;
Do[A[x_] = x + 5 A[4x/5 + A[x]/5]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - F^2) - 4*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 5 * A( 4*x/5 + A(x)/5 )^2.
(2) A(x) = -4*x + 5 * Series_Reversion(x - A(x)^2).
(3) R(x) = -x/4 + 5/4 * Series_Reversion(x + 4*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = x/5 + 4*R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 5^(n-k-1).

A277301 G.f. satisfies: A(x - 2*A(x)^2) = x + 3*A(x)^2.

Original entry on oeis.org

1, 5, 70, 1425, 35410, 999210, 30855820, 1020407105, 35642665050, 1302725802510, 49490450201460, 1944619121474970, 78734794663758580, 3275324221277662900, 139667810517388712600, 6093781146211490413825, 271623891311306597652650, 12353670814537544856558950, 572686428900679117724156900, 27036308383662996662940155550, 1298856469077709523772645582300
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 70*x^3 + 1425*x^4 + 35410*x^5 + 999210*x^6 + 30855820*x^7 + 1020407105*x^8 + 35642665050*x^9 + 1302725802510*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 2*F^2) - 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 5 * A( 3*x/5 + 2*A(x)/5 )^2.
(2) A(x) = -3*x/2 + 5/2 * Series_Reversion(x - 2*A(x)^2).
(3) R(x) = -2*x/3 + 5/3 * Series_Reversion(x + 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = 2*x/5 + 3*R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 2^k * 5^(n-k-1).

A277302 G.f. satisfies: A(x - 3*A(x)^2) = x + 2*A(x)^2.

Original entry on oeis.org

1, 5, 80, 1900, 55490, 1848660, 67630080, 2657251005, 110560510400, 4824793769260, 219334788340040, 10334817935549420, 502814686712631520, 25184673137026274600, 1295595210394570426800, 68326193725188929358600, 3688253200687778850553800, 203524353764195058692833200, 11468618360097679305600299400, 659345494779348103800864088800, 38644445208422874351089132287200
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 80*x^3 + 1900*x^4 + 55490*x^5 + 1848660*x^6 + 67630080*x^7 + 2657251005*x^8 + 110560510400*x^9 + 4824793769260*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277303, A277304, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 3*F^2) - 2*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 5 * A( 2*x/5 + 3*A(x)/5 )^2.
(2) A(x) = -2*x/3 + 5/3 * Series_Reversion(x - 3*A(x)^2).
(3) R(x) = -3*x/2 + 5/2 * Series_Reversion(x + 2*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = 3*x/5 + 2*R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 3^k * 5^(n-k-1).

A277303 G.f. satisfies: A(x - 4*A(x)^2) = x + A(x)^2.

Original entry on oeis.org

1, 5, 90, 2425, 80630, 3065810, 128271540, 5774538945, 275743894750, 13832116773110, 723891526915820, 39323723086794730, 2208811824884144540, 127904686371063157700, 7617441454740093233000, 465691699545009287055825, 29179499379365501297165550, 1871486497257264286902367950, 122731222232573572625823907900, 8222122259910817121846641763950, 562251437460415648354364719018900
Offset: 1

Views

Author

Paul D. Hanna, Oct 09 2016

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 90*x^3 + 2425*x^4 + 80630*x^5 + 3065810*x^6 + 128271540*x^7 + 5774538945*x^8 + 275743894750*x^9 + 13832116773110*x^10 +...

Links

Crossrefs

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.
Cf. A277300, A277301, A277302, A277304, A277305, A277306, A277307, A277308, A277309.
Cf. A276364.

Programs

  • PARI
    {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 4*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) also satisfies:
(1) A(x) = x + 5 * A( x/5 + 4*A(x)/5 )^2.
(2) A(x) = -x/4 + 5/4 * Series_Reversion(x - 4*A(x)^2).
(3) R(x) = -4*x + 5 * Series_Reversion(x + A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x/5 - R(x)/5 ) ) = 4*x/5 + R(x)/5, where R(A(x)) = x.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 4^k * 5^(n-k-1).
Showing 1-10 of 18 results. Next

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