A091200 -id:A091200 - cp's OEIS frontend

A091190 G.f. A(x) satisfies xA(x)^3 = B(xA(x^3)) where B(x) = x/(1 - 3*x).

1, 1, 2, 5, 13, 35, 97, 273, 778, 2240, 6499, 18976, 55703, 164243, 486130, 1443620, 4299365, 12836825, 38413933, 115184282, 346005073, 1041072108, 3137060983, 9465689545, 28596915843, 86492865522, 261876842801, 793661873276

Offset: 0

Paul D. Hanna, Feb 22 2004

nonn

More generally, given A(x) satisfies x*A(x)^p = B(x*A(x^p)) where B(x) = x/(1-p*x), then it appears that A(x) is an integer series only when p is prime. This is a special case of sequences with g.f.s that satisfy the more general functional equation x*A(x)^m = B(x*A(x^m)) studied by Michael Somos; some other examples are A085748, A091188 and A091200.

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + 273*x^7 + 778*x^8 + 2240*x^9 + 6499*x^10 + 18976*x^11 + 55703*x^12 + ...
where A(x)^3 = A(x^3) / (1 - 3*x*A(x^3)).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 87*x^4 + 270*x^5 + 839*x^6 + 2607*x^7 + 8100*x^8 + 25169*x^9 + 78207*x^10 + 243009*x^11 + 755095*x^12 + ...
Also, D(x) = x*A(D(x)) is the g.f. of A370441, which begins
D(x) = x + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 261*x^6 + 1324*x^7 + 6952*x^8 + 37461*x^9 + ... + A370441(n)*x^n + ...
such that D(x)^3 = D( x^3 + 3*D(x)^4 ).

Paul D. Hanna, Table of n, a(n) for n = 0..900

Cf. A085748, A091188, A091200, A370441.

m = 28; B[x_] = x/(1 - 3 x); A[_] = 1;
Do[A[x_] = (B[x A[x^3]]/x)^(1/3) + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 29 2019 *)

{a(n) = my(A,p=3,m=1); if(n<0,0, m=1; A=1+O(x); while(m<=n, m*=p; A = x*subst(A,x,x^p); A = (A/(1-p*A)/x)^(1/p)); polcoeff(A,n))}
for(n=0,30, print1(a(n),", "))

From Paul D. Hanna, Mar 09 2024: (Start)
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A(x^3) / (1 - 3*x*A(x^3)).
(2) A(x) = x/Series_Reversion(D(x)) where D(x) = x*A(D(x)) is the g.f. of A370441.
(End)

Corrected by T. D. Noe, Oct 25 2006

A091188 G.f. A(x) satisfies both A(-x)A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x(1+x)/(1-x).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 10, 12, 23, 31, 58, 79, 145, 207, 374, 540, 964, 1427, 2522, 3775, 6626, 10050, 17532, 26811, 46561, 71795, 124188, 192661, 332228, 518303, 891340, 1396902, 2396912, 3771822, 6459202, 10199912, 17437727, 27622807, 47152952

Offset: 0

Paul D. Hanna, Feb 22 2004

nonn

This is a special case of sequences with g.f.s that satisfy the more general functional equation xA(x)^m = B(xA(x^m)) originated by Michael Somos; some other examples are A085748, A091190 and A091200.

			1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 10*x^7 + 12*x^8 + 23*x^9 + ...
q + q^3 + q^5 + 2*q^7 + 2*q^9 + 4*q^11 + 5*q^13 + 10*q^15 + 12*q^17 + ...

Cf. A085748, A091190, A091200, A092869.

{a(n) = local(A, m); if( n<0, 0, m=1; A = 1 + O(x); while( m<=n, m*=2; A = x * subst(A, x, x^2); A = (A *(1 + A) /(1 - A) / x)^(1/2)); polcoeff(A, n))}

Given g.f. A(x), then B(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^2)) were f(u, v) = u^2 * (1 - v) - v * (1 + v). - Michael Somos, Aug 02 2011

A361764 Expansion of g.f. A(x) satisfying A(x)^5 = A( x^5/(1 - 5x)^5 ) / (1 - 5x).

Original entry on oeis.org

1, 1, 3, 11, 44, 185, 806, 3627, 16926, 82615, 425633, 2325804, 13438568, 81258283, 507109592, 3223435416, 20655599675, 132496854084, 847152571284, 5386490329194, 34026141582719, 213512516149309, 1331393810596499, 8255968489237781, 50955585198416275, 313329163267012645

Offset: 0

Paul D. Hanna, Mar 24 2023

nonn

Related Catalan identity: C(x)^2 = C( x^2/(1 - 2*x)^2 ) / (1 - 2*x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Radius of convergence of g.f. A(x) is r where r is the real root of r = (1 - 5*r)^(5/4) with A(r) = 1/r^(1/5) = 1.451902871451714... so that A(r)^5 = A(r)/(1 - 5*r) and r = 0.1549930338264677513709380922535...

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 44*x^4 + 185*x^5 + 806*x^6 + 3627*x^7 + 16926*x^8 + 82615*x^9 + 425633*x^10 + ...
such that A(x)^5 = A( x^5/(1 - 5*x)^5 ) / (1 - 5*x).
RELATED SERIES.
A(x)^5 = 1 + 5*x + 25*x^2 + 125*x^3 + 625*x^4 + 3126*x^5 + 15655*x^6 + 78650*x^7 + 397625*x^8 + 2031875*x^9 + 10553128*x^10 + ...
A( x^5/(1 - 5*x)^5 ) = 1 + x^5 + 25*x^6 + 375*x^7 + 4375*x^8 + 43750*x^9 + 393753*x^10 + 3281400*x^11 + 25785375*x^12 + ...
SPECIFIC VALUES.
A(1/7) = ( 7/2 * A(1/32) )^(1/5) = 1.293495906485927953020670787280...
A(1/7) = (1 - 5/7)^(-1/5) * (1 - 5/32)^(-1/25) * (1 - 5/14348907)^(-1/125) * (1 - 5/14348902^5)^(-1/625) * ...
A(1/8) = ( 8/3 * A(1/243) )^(1/5) = 1.21774097368643014934892826038499995...
A(1/8) = (1 - 5/8)^(-1/5) * (1 - 5/243)^(-1/25) * (1 - 5/763633171168)^(-1/125) * (1 - 5/763633171163^5)^(-1/625) * ...
A(1/10) = ( 2 * A(1/3125) )^(1/5) = 1.14877193292427434012390599513357372...
A(1/10) = (1 - 5/10)^(-1/5) * (1 - 5/3125)^(-1/25) * (1 - 5/295646655283200000)^(-1/125) * (1 - 5/295646655283199995^5)^(-1/625) * ...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A361765, A361762, A091200, A000108.

{a(n) = my(A=1); for(i=1, #binary(n+1), A = ( subst(A, x, x^5/(1 - 5*x +x*O(x^n))^5 )/(1 - 5*x +x*O(x^n)) )^(1/5) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x)^5 = A( x^5/(1 - 5*x)^5 ) / (1 - 5*x).
(2) A(x^5) = A( x/(1 + 5*x) )^5 / (1 + 5*x).
(3) A(x) = Product_{n>=1} 1/(1 - 5/F(n,x))^(1/5^n), where F(1,x) = 1/x, F(m,x) = (F(m-1,x) - 5)^5 for m > 1.

A370545 Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).

Original entry on oeis.org

1, 1, 4, 21, 125, 801, 5388, 37518, 268109, 1955000, 14487754, 108794169, 826054062, 6331064385, 48914088750, 380555960864, 2978892961194, 23444095375593, 185394136871818, 1472396312841250, 11739089289817538, 93921736129064325, 753845680317416682, 6068255413854119432

Offset: 1

Paul D. Hanna, Mar 26 2024

nonn

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

			G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 125*x^5 + 801*x^6 + 5388*x^7 + 37518*x^8 + 268109*x^9 + 1955000*x^10 + 14487754*x^11 + 108794169*x^12 + ...
where A(x)^5 = A( x^5 + 5*A(x)^6 ).
RELATED SERIES.
A(x)^5 = x^5 + 5*x^6 + 30*x^7 + 195*x^8 + 1330*x^9 + 9376*x^10 + 67720*x^11 + ...
A(x)^6 = x^6 + 6*x^7 + 39*x^8 + 266*x^9 + 1875*x^10 + 13542*x^11 + 99654*x^12 + ...
Let B(x) be the series reversion of A(x), A(B(x)) = x, which begins
B(x) = x - x^2 - 2*x^3 - 6*x^4 - 21*x^5 - 80*x^6 - 320*x^7 - 1326*x^8 - 5637*x^9 - 24434*x^10 - ... + (-1)^(n-1)*A352703(n-1)*x^n + ...
then B(x)^5 + 5*x^6 = B(x^5).
Let C(x) = x^2/B(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 802*x^7 + 3553*x^8 + 15994*x^9 + 72886*x^10 + ... + A091200(n-1)*x^n + ...
where A(x^2/C(x)) = x and C(A(x)) = A(x)^2/x,
then C(x)^5 = C(x^5)/(1 - 5*C(x^5)/x^4).

Paul D. Hanna, Table of n, a(n) for n = 1..400

Cf. A352703, A091200, A271931, A370546.

{a(n) = my(A=x+x^2); for(m=1, n, A=truncate(A); A = subst(A, x, x^5 + 5*A^6 +x^5*O(x^m))^(1/5) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

cp's OEIS Frontend

A091190 G.f. A(x) satisfies x*A(x)^3 = B(x*A(x^3)) where B(x) = x/(1 - 3*x).

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A091188 G.f. A(x) satisfies both A(-x)*A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x*(1+x)/(1-x).

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A361764 Expansion of g.f. A(x) satisfying A(x)^5 = A( x^5/(1 - 5*x)^5 ) / (1 - 5*x).

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A370545 Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).

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A091190 G.f. A(x) satisfies xA(x)^3 = B(xA(x^3)) where B(x) = x/(1 - 3*x).

A091188 G.f. A(x) satisfies both A(-x)A(x) = A(x^2) and xA(x)^2 = B(xA(x^2)) where B(x) = x(1+x)/(1-x).

A361764 Expansion of g.f. A(x) satisfying A(x)^5 = A( x^5/(1 - 5x)^5 ) / (1 - 5x).