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.

A368629 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^4 - A(-x)^4)/2.

Original entry on oeis.org

1, 1, 4, 9, 88, 210, 2644, 6493, 91992, 229646, 3484008, 8789562, 139443168, 354379540, 5801987316, 14824740645, 248459660984, 637465292438, 10878564788984, 28001827694446, 484778825103504, 1251132971284668, 21915195896364296, 56682787977509650, 1002570518541796720
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2024

Keywords

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 88*x^4 + 210*x^5 + 2644*x^6 + 6493*x^7 + 91992*x^8 + 229646*x^9 + 3484008*x^10 + 8789562*x^11 + 139443168*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 210*x^4 + 668*x^5 + 6493*x^6 + 21538*x^7 + 229646*x^8 + 779772*x^9 + 8789562*x^10 + ...
A(x)^4 = 1 + 4*x + 22*x^2 + 88*x^3 + 605*x^4 + 2644*x^5 + 20114*x^6 + 91992*x^7 + 741154*x^8 + 3484008*x^9 + 29125100*x^10 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 9*x^3 + 210*x^5 + 6493*x^7 + 229646*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 9*x^2 + 210*x^4 + 6493*x^6 + 229646*x^8 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^4:
(A(x) + A(-x))/2 = 1 + 4*x^2 + 88*x^4 + 2644*x^6 + 91992*x^8 + 3484008*x^10 + ...
(A(x)^4 - A(-x)^4)/2 = 4*x + 88*x^3 + 2644*x^5 + 91992*x^7 + 3484008*x^9 + ...
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(2) at r = (sqrt(2) - 1)/3 = 0.138071....

Links

Crossrefs

Cf. A368633, A368628, A368593, A368626, A368627.

Programs

  • PARI
    {a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 + B^2)/2 + x*(A^4 - B^4)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^4 - A(-x)^4)/2.
(2) A(x) = A(-x) + x*A(x)^2 + x*A(-x)^2.
(3) A(x) = 2 - A(-x) + x*A(x)^4 - x*A(-x)^4.
(4) A(x) = 2 - A(-x) + (A(x) - A(-x))*(A(x)^2 - A(-x)^2).
(5.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
(5.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
(6) (A(x) + A(-x))/2 = 1/(1 - (A(x) - A(-x))^2).
(7.a) Sum_{n>=0} a(n) * (sqrt(2) - 1)^n/3^n = sqrt(2).
(7.b) Sum_{n>=0} a(n) * (1 - sqrt(2))^n/3^n = 1.

A368633 Expansion of g.f. A(x) satisfying A(x) = 1 + 2*x*A(x)^2 - x*A(-x)^2.

Original entry on oeis.org

1, 1, 6, 13, 114, 290, 2892, 7901, 84090, 239222, 2648244, 7732914, 87894324, 261371940, 3027588120, 9125058525, 107215635402, 326501869166, 3879094785060, 11910103389734, 142766337272988, 441265565242268, 5328172865489448, 16559430499708018, 201171901999797924
Offset: 0

Views

Author

Paul D. Hanna, Jan 11 2024

Keywords

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0 and even elsewhere.

Examples

			G.f. A(x) = 1 + x + 6*x^2 + 13*x^3 + 114*x^4 + 290*x^5 + 2892*x^6 + 7901*x^7 + 84090*x^8 + 239222*x^9 + 2648244*x^10 + 7732914*x^11 + 87894324*x^12 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 13*x^2 + 38*x^3 + 290*x^4 + 964*x^5 + 7901*x^6 + 28030*x^7 + 239222*x^8 + 882748*x^9 + 7732914*x^10 + 29298108*x^11 + 261371940*x^12 + ...
that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 13*x^3 + 290*x^5 + 7901*x^7 + 239222*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 13*x^2 + 290*x^4 + 7901*x^6 + 239222*x^8 + ...
and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 6*x^2 + 114*x^4 + 2892*x^6 + 84090*x^8 + 2648244*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 38*x^3 + 964*x^5 + 28030*x^7 + 882748*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + 3*x * (A(x)^2 - A(-x)^2)/2.
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(2) at r = (sqrt(2) - 1)/3 = 0.138071187457698....

Links

Crossrefs

Cf. A368634, A368629, A368627.

Programs

  • PARI
    {a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A,x,-x);
A = 1 + x*(A^2 + B^2)/2 + 3*x*(A^2 - B^2)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A); A = (1/x)*serreverse( (1 + 4*x - 6*x^2 - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)))/2 ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = 1 + 2*x*A(x)^2 - x*A(-x)^2.
(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 3*x*(A(x)^2 - A(-x)^2)/2.
(2.a) (A(x) + A(-x))/2 = 1 + 3*x*(A(x)^2 - A(-x)^2)/2.
(2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
(2.c) (A(x) + A(-x))/2 = 1/(1 - 3*x*(A(x) - A(-x))).
(3.a) A(x) = (1 - sqrt(1-8*x + 8*x^2*A(-x)^2)) / (4*x).
(3.b) A(-x) = (sqrt(1+8*x + 8*x^2*A(x)^2) - 1) / (4*x).
(4.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
(5) 0 = (1-x) - (1-4*x)*A(x) - 2*x*(1+3*x)*A(x)^2 + 12*x^2*A(x)^3 - 9*x^3*A(x)^4.
(6) x = (1 + 4*x*A(x) - 6*x^2*A(x)^2 - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2))/2.
(7) A(x) = (1/x)*Series_Reversion( (1 + 4*x - 6*x^2 - sqrt(1 + 4*x - 4*x^2))/2 ).
(8.a) Sum_{n>=0} a(n) * (sqrt(2) - 1)^n/3^n = sqrt(2).
(8.b) Sum_{n>=0} a(n) * (1 - sqrt(2))^n/3^n = 1.
Conjecture D-finite with recurrence +375*n*(127034*n -380695)*(n-1)*(n+1) *a(n) -50*n*(n-1) *(1761680*n^2 -13025595*n +22423399)*a(n-1) -24*(n-1) *(85874984*n^3 -429099788*n^2 +603573743*n -272338575)*a(n-2) +8*(476358272*n^4 -5427553976*n^3 +22342502584*n^2 -39872302249*n +26255347914)*a(n-3) +864*(2*n-7) *(508136*n^3 -2793120*n^2 +5307886*n -3864543)*a(n-4) -1152*(n-4) *(352336*n-843439) *(2*n-7) *(2*n-9)*a(n-5)=0. - R. J. Mathar, Jan 24 2024

A368626 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^3 + A(-x)^3)/2.

Original entry on oeis.org

1, 1, 2, 9, 22, 138, 356, 2585, 6830, 53838, 144156, 1197546, 3233692, 27859444, 75665736, 669553209, 1826204958, 16493851110, 45131989100, 414263198030, 1136416283860, 10568504182860, 29050963193720, 273107307342090, 751985844723308, 7133921326564172, 19670502565821464
Offset: 0

Views

Author

Paul D. Hanna, Jan 09 2024

Keywords

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0 and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 22*x^4 + 138*x^5 + 356*x^6 + 2585*x^7 + 6830*x^8 + 53838*x^9 + 144156*x^10 + 1197546*x^11 + 3233692*x^12 + ...
where A(x) is formed from the odd bisection of A(x)^2 and the even bisection of A(x)^3, as can be seen from the expansions
A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 66*x^4 + 356*x^5 + 1157*x^6 + 6830*x^7 + 23222*x^8 + 144156*x^9 + 504546*x^10 + ...
A(x)^3 = 1 + 3*x + 9*x^2 + 40*x^3 + 138*x^4 + 693*x^5 + 2585*x^6 + 13764*x^7 + 53838*x^8 + 296646*x^9 + 1197546*x^10 + ...
so that the bisections of the above series are related by
(A(x) + A(-x))/2 = 1 + x*(A(x)^2 - A(-x)^2)/2, and
(A(x) - A(-x))/2 = x*(A(x)^3 + A(-x)^3)/2.
SPECIFIC VALUES.
A(t) = 3/2 at t = 0.1819737010113140094420890735437063355509087658723835...
with A(-t) = 0.7945570310255352575261389299040205708629421553742768...
G.f. A(x) diverges at x = 1/5.4, but converges at x = 1/5.5 to yield
A(1/5.5) = 1.496543384376249917206500686071412596234401473798923...
A(-1/5.5) = 0.795582249398671834477410218197255634423553817319574...
Other values are as follows.
A(1/6) = 1.34228124014121938629204994980825043322418782558714594...
A(-1/6) = 0.84031658679173656850293071643280362490543801455743768...
A(1/7) = 1.23812032178413019856840253750104622400159644919325618...
A(-1/7) = 0.87219621912499007272745977375746581998964690903627574...
A(1/8) = 1.18723993315598647777707954645984780429075497185978705...
A(-1/8) = 0.88995083754758616465388572384122362483578619460668827...

Links

Crossrefs

Cf. A368627.

Programs

  • PARI
    {a(n) = my(A=1+x, A_); for(i=1, n, A=truncate(A) + x*O(x^i); B=subst(A,x,-x); A = 1 + x*(A^2 - B^2)/2 + x*(A^3 + B^3)/2 ; ); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^3 + A(-x)^3)/2.
(2) A(x) = 2 - A(-x) + x*A(x)^2 - x*A(-x)^2.
(3) A(x) = A(-x) + x*A(x)^3 + x*A(-x)^3.
(4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x).
(5) (A(x) + A(-x))/2 = 1/(1 - 2*x*(A(x) - A(-x))/2).

A368634 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(3*A(x)^2 - A(-x)^2)/2.

Original entry on oeis.org

1, 1, 4, 9, 52, 138, 904, 2581, 18020, 53622, 389112, 1189146, 8855560, 27571156, 209174544, 660249549, 5079702852, 16203796158, 126033559960, 405408758062, 3180991167640, 10301855821452, 81414086371696, 265150389430914, 2108026107021224, 6897985805906972, 55119920086104496
Offset: 0

Views

Author

Paul D. Hanna, Jan 12 2024

Keywords

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0 and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 52*x^4 + 138*x^5 + 904*x^6 + 2581*x^7 + 18020*x^8 + 53622*x^9 + 389112*x^10 + 1189146*x^11 + 8855560*x^12 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 138*x^4 + 452*x^5 + 2581*x^6 + 9010*x^7 + 53622*x^8 + 194556*x^9 + 1189146*x^10 + 4427780*x^11 + 27571156*x^12 + ...
that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 9*x^3 + 138*x^5 + 2581*x^7 + 53622*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 9*x^2 + 138*x^4 + 2581*x^6 + 53622*x^8 + ...
and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 4*x^2 + 52*x^4 + 904*x^6 + 18020*x^8 + 389112*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 26*x^3 + 452*x^5 + 9010*x^7 + 194556*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + 2*x * (A(x)^2 - A(-x)^2)/2.
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(3) at r = (sqrt(3) - 1)/4 = 0.183012701892219....

Links

Crossrefs

Cf. A368633, A368629, A368627.

Programs

  • PARI
    {a(n) = my(A=1+x, B); for(i=1, n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 + B^2)/2 + 2*x*(A^2 - B^2)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A); A = (1/x)*serreverse( (1 + 6*x - 8*x^2 - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)))/4 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = 1 + x*(3*A(x)^2 - A(-x)^2)/2.
(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 2*x*(A(x)^2 - A(-x)^2)/2.
(2.a) (A(x) + A(-x))/2 = 1 + 2*x*(A(x)^2 - A(-x)^2)/2.
(2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
(2.c) (A(x) + A(-x))/2 = 1/(1 - 2*x*(A(x) - A(-x))).
(3.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
(3.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
(4.a) A(x) = (1 - sqrt(1-16*x + 8*x*A(-x) + 16*x^2*A(-x)^2)) / (4*x).
(4.b) A(-x) = (sqrt(1+16*x - 8*x*A(x) + 16*x^2*A(x)^2) - 1) / (4*x).
(5) 0 = (1-2*x) - (1-6*x)*A(x) - x*(3+8*x)*A(x)^2 + 12*x^2*A(x)^3 - 8*x^3*A(x)^4.
(6) x = (1 + 6*x*A(x) - 8*x^2*A(x)^2 - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2))/4.
(7) A(x) = (1/x)*Series_Reversion( (1 + 6*x - 8*x^2 - sqrt(1 + 4*x - 4*x^2))/4 ).
(8.a) Sum_{n>=0} a(n) * (sqrt(3) - 1)^n/4^n = sqrt(3).
(8.b) Sum_{n>=0} a(n) * (1 - sqrt(3))^n/4^n = 1.

A368628 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^4 + A(-x)^4)/2.

Original entry on oeis.org

1, 1, 2, 14, 32, 345, 810, 10492, 24880, 356252, 848992, 12946094, 30942208, 492621678, 1179648292, 19379467704, 46468665184, 781821568212, 1876521420624, 32169136799832, 77270414837888, 1344812759618473, 3232175494812466, 56957048059132524, 136958995341531504
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2024

Keywords

Comments

Conjecture: a(n) is odd when n = (4^k - 1)/3 for k >= 0, and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 32*x^4 + 345*x^5 + 810*x^6 + 10492*x^7 + 24880*x^8 + 356252*x^9 + 848992*x^10 + 12946094*x^11 + 30942208*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 32*x^3 + 96*x^4 + 810*x^5 + 2634*x^6 + 24880*x^7 + 84668*x^8 + 848992*x^9 + 2974649*x^10 + ...
A(x)^4 = 1 + 4*x + 14*x^2 + 84*x^3 + 345*x^4 + 2324*x^5 + 10492*x^6 + 74540*x^7 + 356252*x^8 + 2609552*x^9 + 12946094*x^10 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 2*x^2 + 32*x^4 + 810*x^6 + 24880*x^8 + 848992*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 32*x^3 + 810*x^5 + 24880*x^7 + 848992*x^9 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^4:
(A(x) - A(-x))/2 = x + 14*x^3 + 345*x^5 + 10492*x^7 + 356252*x^9 + ...
(A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 345*x^4 + 10492*x^6 + 356252*x^8 + ...

Links

Crossrefs

Cf. A368629, A368593, A368626, A368627.

Programs

  • PARI
    {a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 - B^2)/2 + x*(A^4 + B^4)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^4 + A(-x)^4)/2.
(2) A(x) = 2 - A(-x) + x*A(x)^2 - x*A(-x)^2.
(3) A(x) = A(-x) + x*A(x)^4 + x*A(-x)^4.
(4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x).
(5) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))).

A368593 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*sqrt( (A(x)^4 + A(-x)^4)/2 ).

Original entry on oeis.org

1, 1, 2, 7, 18, 78, 220, 1043, 3090, 15402, 47044, 242126, 755076, 3973820, 12580344, 67303139, 215511330, 1167556434, 3772175860, 20640707866, 67167649868, 370510806212, 1212836703304, 6735128062542, 22156120392276, 123731147310820, 408741630687656, 2293595176625340
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2024

Keywords

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 18*x^4 + 78*x^5 + 220*x^6 + 1043*x^7 + 3090*x^8 + 15402*x^9 + 47044*x^10 + 242126*x^11 + 755076*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 54*x^4 + 220*x^5 + 717*x^6 + 3090*x^7 + 10562*x^8 + 47044*x^9 + 165858*x^10 + 755076*x^11 + ...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 205*x^4 + 836*x^5 + 3178*x^6 + 13192*x^7 + 51490*x^8 + 216808*x^9 + 862588*x^10 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 2*x^2 + 18*x^4 + 220*x^6 + 3090*x^8 + 47044*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 18*x^3 + 220*x^5 + 3090*x^7 + 47044*x^9 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^4:
(A(x) - A(-x))/2 = x + 7*x^3 + 78*x^5 + 1043*x^7 + 15402*x^9 + ...
(A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 205*x^4 + 3178*x^6 + 51490*x^8 + ...
sqrt( (A(x)^4 + A(-x)^4)/2 ) = 1 + 7*x^2 + 78*x^4 + 1043*x^6 + 15402*x^8 + ...

Links

Crossrefs

Cf. A368628, A368629, A368626, A368627.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n,
A = Vec(1 + x*(Ser(A)^2 - subst(Ser(A)^2,x,-x))/2 + x*sqrt( (Ser(A)^4 + subst(Ser(A)^4,x,-x))/2 ) +x*O(x^#A) ) );A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*sqrt( (A(x)^4 + A(-x)^4)/2 ).
(2) A(x) = 2 - A(-x) + x*A(x)^2 - x*A(-x)^2.
(3) A(x) = A(-x) + x*sqrt( (A(x)^4 + x*A(-x)^4)/2 ).
(4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x).
(5) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))).

A368635 Expansion of g.f. A(x) satisfying A(x) = 1 + 3*x*A(x)^2 - 2*x*A(-x)^2.

Original entry on oeis.org

1, 1, 10, 21, 310, 762, 12820, 33805, 607550, 1667214, 31182540, 87799362, 1686609820, 4835044372, 94676506920, 275037241149, 5463738069390, 16035014605830, 322140216214300, 953095126595062, 19320606147948820, 57539265876939756, 1175037853461723160, 3518503980453113106
Offset: 0

Views

Author

Paul D. Hanna, Jan 12 2024

Keywords

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 10*x^2 + 21*x^3 + 310*x^4 + 762*x^5 + 12820*x^6 + 33805*x^7 + 607550*x^8 + 1667214*x^9 + 31182540*x^10 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 21*x^2 + 62*x^3 + 762*x^4 + 2564*x^5 + 33805*x^6 + 121510*x^7 + 1667214*x^8 + 6236508*x^9 + 87799362*x^10 + ...
that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 21*x^3 + 762*x^5 + 33805*x^7 + 1667214*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 21*x^2 + 762*x^4 + 33805*x^6 + 1667214*x^8 + ...
and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 10*x^2 + 310*x^4 + 12820*x^6 + 607550*x^8 + 31182540*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 62*x^3 + 2564*x^5 + 121510*x^7 + 6236508*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + 5*x * (A(x)^2 - A(-x)^2)/2.
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(6)/2 at r = (sqrt(6) - 2)/5 = 0.0898979485566356....

Links

Crossrefs

Cf. A368633, A368634, A368629, A368627.

Programs

  • PARI
    {a(n) = my(A=1+x, B); for(i=1, n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 + B^2)/2 + 5*x*(A^2 - B^2)/2 ); polcoeff(A, n)}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(A); A = (1/x)*serreverse( (1 + 3*x - 5*x^2) - sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = 1 + 3*x*A(x)^2 - 2*x*A(-x)^2.
(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + 5*x*(A(x)^2 - A(-x)^2)/2.
(2.a) (A(x) + A(-x))/2 = 1 + 5*x*(A(x)^2 - A(-x)^2)/2.
(2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
(2.c) (A(x) + A(-x))/2 = 1/(1 - 5*x*(A(x) - A(-x))).
(3.a) A(x) = (1 - sqrt(1 - 40*x + 20*x*A(-x) + 100*x^2*A(-x)^2))/(10*x).
(3.b) A(-x) = (sqrt(1 + 40*x - 20*x*A(x) + 100*x^2*A(x)^2) - 1)/(10*x).
(4.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2))/(2*x).
(4.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
(5) 0 = (2 - x) - 2*(1-3*x)*A(x) - x*(3+10*x)*A(x)^2 + 30*x^2*A(x)^3 - 25*x^3*A(x)^4.
(6) x = (1 + 3*x*A(x) - 5*x^2*A(x)^2) - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2).
(7) A(x) = (1/x)*Series_Reversion( (1 + 3*x - 5*x^2) - sqrt(1 + 4*x - 4*x^2) ).
(8.a) Sum_{n>=0} a(n) * (sqrt(6) - 2)^n/5^n = sqrt(6)/2.
(8.b) Sum_{n>=0} a(n) * (2 - sqrt(6))^n/5^n = 1.
D-finite with recurrence +10935*n*(n-1)*(75132*n-217883) *(n+1)*a(n) -1458*n*(n-1) *(865728*n^2 -6402143*n+11031849)*a(n-1) -360 *(n-1)*(159580368*n^3 -781944228*n^2 +1079436906*n -451430219)*a(n-2) +24*(3677612544*n^4 -41906753640*n^3 +172755991440*n^2 -308392913875*n +202512185406)*a(n-3) +12000*(2*n-7) *(300528*n^3 -1622852*n^2 +3317670*n -2937917)*a(n-4) -3200*(n-4)*(865728*n -2073503) *(2*n-7)*(2*n-9) *a(n-5)=0. - R. J. Mathar, Jan 25 2024

A369083 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(5*A(x)^2 - A(-x)^2)/4.

Original entry on oeis.org

1, 1, 3, 7, 30, 83, 402, 1199, 6180, 19232, 102939, 329217, 1807344, 5891442, 32936724, 108884607, 617125788, 2062285676, 11813994060, 39818644316, 230067933810, 780838528379, 4543410985386, 15509003672617, 90771938228244, 311354249554852, 1831389290870538, 6307784087296006
Offset: 0

Views

Author

Paul D. Hanna, Jan 12 2024

Keywords

Comments

Conjecture: a(n) == binomial(4*n+3,n) (mod 2) for n >= 0 (cf. A263133).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 30*x^4 + 83*x^5 + 402*x^6 + 1199*x^7 + 6180*x^8 + 19232*x^9 + 102939*x^10 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 83*x^4 + 268*x^5 + 1199*x^6 + 4120*x^7 + 19232*x^8 + 68626*x^9 + 329217*x^10 + ...
that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 7*x^3 + 83*x^5 + 1199*x^7 + 19232*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 7*x^2 + 83*x^4 + 1199*x^6 + 19232*x^8 + ...
and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 3*x^2 + 30*x^4 + 402*x^6 + 6180*x^8 + 102939*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 20*x^3 + 268*x^5 + 4120*x^7 + 68626*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + (3/2)*x * (A(x)^2 - A(-x)^2)/2.

Links

Crossrefs

Cf. A368633, A368634, A368635, A368627, A368629.

Programs

  • PARI
    {a(n) = my(A=1+x, B); for(i=1, n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 + B^2)/2 + (3/2)*x*(A^2 - B^2)/2 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A); A = (1/x)*serreverse( (1 + 10*x - 12*x^2 - sqrt(1 + 4*x - 4*x^2  +x^2*O(x^n) ))/8 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = 1 + x*(5*A(x)^2 - A(-x)^2)/4.
(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + (3/2)*x*(A(x)^2 - A(-x)^2)/2.
(2.a) (A(x) + A(-x))/2 = 1 + (3/2)*x*(A(x)^2 - A(-x)^2)/2.
(2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
(2.c) (A(x) + A(-x))/2 = 1/(1 - 3*x*(A(x) - A(-x))/2).
(3.a) A(x) = (1 - sqrt(1 - 12*x + 6*x*A(-x) + 9*x^2*A(-x)^2)) / (3*x).
(3.b) A(-x) = (sqrt(1 + 12*x - 6*x*A(x) + 9*x^2*A(x)^2) - 1) / (3*x).
(4.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
(5) 0 = (1-4*x) - (1-10*x)*A(x) - (5+12*x)*x*A(x)^2 + 15*x^2*A(x)^3 - 9*x^3*A(x)^4.
(6) x = (1 + 10*x*A(x) - 12*x^2*A(x)^2 - sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2))/8.
(7) A(x) = (1/x)*Series_Reversion( (1 + 10*x - 12*x^2 - sqrt(1 + 4*x - 4*x^2))/8 ).

A369082 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(3*A(x)^2 + A(-x)^2)/4.

Original entry on oeis.org

1, 1, 1, 3, 4, 15, 22, 91, 140, 612, 969, 4389, 7084, 32890, 53820, 254475, 420732, 2017356, 3362260, 16301164, 27343888, 133767543, 225568798, 1111731933, 1882933364, 9338434700, 15875338990, 79155435870, 134993766600, 676196049060, 1156393243320, 5815796869995
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2024

Keywords

Comments

Equals the interleaving of sequences A002293 and A006632.

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 4*x^4 + 15*x^5 + 22*x^6 + 91*x^7 + 140*x^8 + 612*x^9 + 969*x^10 + 4389*x^11 + 7084*x^12 + ...
RELATED SERIES.
We can see from the expansion of A(x)^2, which begins
A(x)^2 = 1 + 2*x + 3*x^2 + 8*x^3 + 15*x^4 + 44*x^5 + 91*x^6 + 280*x^7 + 612*x^8 + 1938*x^9 + 4389*x^10 + 14168*x^11 + 32890*x^12 + ...
that the odd bisection of A(x) is derived from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 3*x^3 + 15*x^5 + 91*x^7 + 612*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 3*x^2 + 15*x^4 + 91*x^6 + 612*x^8 + ...
and the even bisection of A(x) is derived from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + x^2 + 4*x^4 + 22*x^6 + 140*x^8 + 969*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 8*x^3 + 44*x^5 + 280*x^7 + 1938*x^9 + ...
so that (A(x) + A(-x))/2 = 1 + (1/2)*x * (A(x)^2 - A(-x)^2)/2.

Links

Crossrefs

Cf. A369083, A368633, A368634, A368635, A368627, A368629.
Cf. A000108, A001764, A002293, A006632.

Programs

  • PARI
    {a(n) = if(n%2==0, binomial(2*n, n/2)/(3*n/2 + 1), 3*binomial(2*n+1,n\2)/(2*n+1))}
for(n=0,30, print1(a(n),", "))
  • PARI
    {a(n) = my(A=1+x, B); for(i=1, n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 + B^2)/2 + (1/2)*x*(A^2 - B^2)/2 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A); A = (1/x)*serreverse( (sqrt(1 + 4*x - 4*x^2 +x^2*O(x^n)) - (1 - 6*x + 4*x^2))/8 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1.a) A(x) = 1 + x*(3*A(x)^2 + A(-x)^2)/4.
(1.b) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + (1/2)*x*(A(x)^2 - A(-x)^2)/2.
(2.a) (A(x) + A(-x))/2 = 1 + (1/2)*x*(A(x)^2 - A(-x)^2)/2.
(2.b) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2.
(2.c) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))/2).
(2.d) (A(x) + A(-x))/2 = F(x^2) where F(x) = 1 + x*F(x)^4 (cf. A002293).
(2.e) (A(x) - A(-x))/2 = x*F(x^2)^3 where F(x) = 1 + x*F(x)^4 (cf. A006632).
(3.a) A(x) = (1 - sqrt(1 - 4*x + 2*x*A(-x) + x^2*A(-x)^2))/x.
(3.b) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2))/(2*x).
(4.a) 0 = (1+4*x) - (1+6*x)*A(x) + (3+4*x)*x*A(x)^2 - 3*x^2*A(x)^3 + x^3*A(x)^4.
(4.b) x = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - (1 - 6*x*A(x) + 4*x^2*A(x)^2))/8.
(5.a) A(x) = (1/x)*Series_Reversion( (sqrt(1 + 4*x - 4*x^2) - (1 - 6*x + 4*x^2))/8 ).
(5.b) (A(x) + A(-x))/2 = (1/x)*Series_Reversion( x/C(x^2) ) = where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan sequence (A000108).
(5.c) (A(x) - A(-x))/2 = Series_Reversion( x*D(-x^2)^3 ) where D(x) = 1 + x*D(x)^3 (cf. A001764).
(6.a) a(2*n) = binomial(4*n, n)/(3*n + 1) for n >= 0.
(6.b) a(2*n+1) = 3*binomial(4*n+3,n)/(4*n+3) for n >= 0.
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