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 1-3 of 3 results.

A264228 G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-3*x) ), with A(0) = 0.

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 97, 274, 785, 2275, 6656, 19630, 58295, 174175, 523238, 1579584, 4789919, 14584723, 44577799, 136732988, 420784888, 1298937282, 4021383654, 12483820395, 38853994422, 121220646116, 379062880051, 1187912517953, 3730305167438, 11736596024002, 36994041916973, 116807229667919, 369415244627269, 1170113816365089
Offset: 1

Views

Author

Paul D. Hanna, Nov 08 2015

Keywords

Comments

Radius of convergence is r = (sqrt(13) - 3)/2, where r = r^3/(1-3*r), with A(r) = 1.
Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 + 58295*x^13 + 174175*x^14 + ...
where A(x)^3 = A( x^3/(1-3*x) ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 87*x^7 + 270*x^8 + 839*x^9 + 2610*x^10 + 8127*x^11 + 25331*x^12 + 79035*x^13 + 246852*x^14 + 771808*x^15 + ...
A( x/(1+x+x^2) ) = x + x^4 + 2*x^7 + 6*x^10 + 22*x^13 + 88*x^16 + 367*x^19 + 1570*x^22 + 6843*x^25 + 30271*x^28 + 135530*x^31 + 612852*x^34 + 2794412*x^37 + 12832472*x^40 + ...
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + x + x^2 + x^3 - x^5 - x^6 + 2*x^8 + 3*x^9 - 6*x^11 - 9*x^12 + 20*x^14 + 30*x^15 - 71*x^17 - 110*x^18 + 267*x^20 + 419*x^21 - 1041*x^23 + ...
Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + x + C2(x):
C0(x) = 1 + x^3 - x^6 + 3*x^9 - 9*x^12 + 30*x^15 - 110*x^18 + 419*x^21 - 1648*x^24 + 6652*x^27 - 27369*x^30 + 114384*x^33 - 484276*x^36 + ...
C2(x) = x^2 - x^5 + 2*x^8 - 6*x^11 + 20*x^14 - 71*x^17 + 267*x^20 - 1041*x^23 + 4168*x^26 - 17047*x^29 + 70902*x^32 + ... + (-1)^(n-1)*A370446(n)*x^(3*n-1) + ...
then C0(x) = x^2/C2(x).

Links

Crossrefs

Cf. A264229, A264230, A361763, A370440, A370446.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3/(1-3*x +x*O(x^n))) )^(1/3) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x)^3 = A( x^3/(1-3*x) ).
(2) A( x/(1+3*x) )^3 = A( x^3/(1+3*x)^2 ). - Paul D. Hanna, Mar 25 2023
(3) A( x/(1+x+x^2) )^3 = A( x^3/(1-x^3)^2 ). - Paul D. Hanna, Mar 11 2024

A361763 Expansion of g.f. A(x) satisfying A(x)^3 = A( x^3/(1 - 3*x)^3 ).

Original entry on oeis.org

1, 3, 9, 28, 93, 333, 1271, 5064, 20673, 85460, 355659, 1486719, 6238608, 26278281, 111114558, 471608944, 2008906581, 8586410085, 36816550550, 158332335279, 682843960665, 2952865525730, 12802463157570, 55646477022330, 242465061290160, 1059022767175173, 4636452916770489
Offset: 1

Views

Author

Paul D. Hanna, Mar 23 2023

Keywords

Comments

Related Catalan identity: F(x)^2 = F( x^2/(1 - 2*x)^2 ), where F(x) = x*C(x)^2 and 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 - 3*r)^(3/2) with A(r) = 1 and r = (52 - (324*sqrt(717) + 8108)^(1/3) + (324*sqrt(717) - 8108)^(1/3))/162 = 0.214054846272632706742...

Examples

			G.f.: A(x) = x + 3*x^2 + 9*x^3 + 28*x^4 + 93*x^5 + 333*x^6 + 1271*x^7 + 5064*x^8 + 20673*x^9 + 85460*x^10 + 355659*x^11 + 1486719*x^12 + ...
where
A( x^3/(1 - 3*x)^3 ) = x^3 + 9*x^4 + 54*x^5 + 273*x^6 + 1269*x^7 + 5670*x^8 + 24957*x^9 + 109593*x^10 + 482598*x^11 + 2133082*x^12 + ...
which equals A(x)^3.
RELATED SERIES.
Notice that the following cube root is an integer series
( A(x)/x )^(1/3) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 197*x^6 + 779*x^7 + 3135*x^8 + 12709*x^9 + 51757*x^10 + ... + A361762(n)*x^n + ...
Also, let B(x) satisfy A(x/B(x)) = x and B(A(x)) = A(x)/x,
then B(x) = x/Series_Reversion(A(x)) is the g.f. of A107092,
B(x) = 1 + 3*x + x^3 - x^6 + 2*x^9 - 4*x^12 + 9*x^15 - 22*x^18 + 55*x^21 - 142*x^24 + 376*x^27 - 1011*x^30 + ...
such that B(x)^3 = B(x^3) + 3*x,
as shown by the series
B(x)^(1/3) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 22*x^6 + 55*x^7 - 142*x^8 + 376*x^9 - 1011*x^10 + ...
SPECIFIC VALUES.
A(1/5) = A(1/8)^(1/3) = 0.586384210523490911367880492498...
A(1/5) = (1/5) * (1 - 3/5)^(-1) * (1 - 3/8)^(-1/3) * (1 - 3/125)^(-1/9) * (1 - 3/1815848)^(-1/27) * ...
A(1/6) = A(1/27)^(1/3) = 0.346688997573685318336777346240...
A(1/6) = (1/6) * (1 - 3/6)^(-1) * (1 - 3/27)^(-1/3) * (1 - 3/13824)^(-1/9) * (1 - 3/2640087986661)^(-1/27) * ...
A(1/9) = A(1/216)^(1/3) = 0.16744549995321182031691216552466...
A(1/12) = A(1/729)^(1/3) = 0.11126394649161862248626102306202...

Links

Crossrefs

Cf. A361762 ((A(x)/x)^(1/3)), A264230, A107092, A091190, A361765.
Cf. A264228, A264229, A264230.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1), A = ( subst(A, x, x^3/(1 - 3*x +x*O(x^n))^3 ) )^(1/3) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x)^3 = A( x^3/(1 - 3*x)^3 ).
(2) A(x^3) = A( x/(1 + 3*x) )^3.
(3) A(x) = x * Product_{n>=0} 1/(1 - 3/F(n,x))^(1/3^n), where F(0,x) = 1/x, F(m,x) = (F(m-1,x) - 3)^3 for m > 0.
(4) x/Series_Reversion(A(x)) = B(x) such that B(x)^3 = B(x^3) + 3*x (cf. A107092).

A264229 G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-6*x) ), with A(0) = 0.

Original entry on oeis.org

1, 2, 8, 38, 192, 1008, 5428, 29752, 165232, 926986, 5242696, 29846440, 170846760, 982496400, 5672562432, 32864292248, 190977464576, 1112761458944, 6499186961080, 38040656888144, 223089977217248, 1310627164161296, 7712227735497024, 45449101195872960, 268204421736352320, 1584740639910023552, 9374834857254623744, 55519826063209918038
Offset: 1

Views

Author

Paul D. Hanna, Nov 08 2015

Keywords

Comments

Radius of convergence is r = (sqrt(40) - 6)/2, where r = r^3/(1-6*r), with A(r) = 1.
Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).

Examples

			G.f.: A(x) = x + 2*x^2 + 8*x^3 + 38*x^4 + 192*x^5 + 1008*x^6 + 5428*x^7 + 29752*x^8 + 165232*x^9 + 926986*x^10 + 5242696*x^11 + 29846440*x^12 + ...
where A(x)^3 = A( x^3/(1-6*x) ).
RELATED SERIES.
A(x)^3 = x^3 + 6*x^4 + 36*x^5 + 218*x^6 + 1320*x^7 + 7992*x^8 + 48392*x^9 + 293040*x^10 + 1774656*x^11 + 10748198*x^12 + 65101584*x^13 + ...
A( x/(1 + 2*x + 4*x^2) ) = x + 6*x^4 + 52*x^7 + 554*x^10 + 6888*x^13 + 95768*x^16 + 1435832*x^19 + 22605648*x^22 + 367354432*x^25 + 6097422934*x^28 + 102720725488*x^31 + 1749623396240*x^34 + 30056679361984*x^37 + ...
Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where
B(x) = 1 + 2*x + 4*x^2 + 6*x^3 - 24*x^5 - 56*x^6 + 368*x^8 + 986*x^9 - 7496*x^11 - 21144*x^12 + 173824*x^14 + 505040*x^15 - 4353184*x^17 + ...
Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + 2*x + C2(x):
C0(x) = 1 + 6*x^3 - 56*x^6 + 986*x^9 - 21144*x^12 + 505040*x^15 - 12892588*x^18 + 344317272*x^21 - 9501257152*x^24 + ...
C2(x) = 4*x^2 - 24*x^5 + 368*x^8 - 7496*x^11 + 173824*x^14 - 4353184*x^17 + 114716608*x^20 - 3134509760*x^23 + ...
then C0(x) = 4*x^2/C2(x).

Crossrefs

Cf. A264228, A264230.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3/(1-6*x +x*O(x^n))) )^(1/3) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

From Paul D. Hanna, Mar 17 2024: (Start)
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x)^3 = A( x^3/(1 - 6*x) ).
(2) A( x/(1 + 6*x) )^3 = A( x^3/(1 + 6*x)^2 ).
(3) A( x/(1 + 2*x + 4*x^2) )^3 = A( x^3/(1 - 8*x^3)^2 ). (End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.