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 1-2 of 2 results.

A274484 G.f. satisfies: A(x)^2 = A( x^2/(1 - 4*x + 2*x^2) ).

Original entry on oeis.org

1, 2, 6, 20, 71, 262, 994, 3852, 15183, 60686, 245410, 1002300, 4128448, 17129920, 71529800, 300355184, 1267386163, 5371101382, 22850230642, 97546995260, 417717017392, 1793765580704, 7722405668232, 33323153856880, 144099312039391, 624347587536782, 2710036186345914, 11782865084403212, 51310167663855675, 223762749750806942, 977155903597684074, 4272633455348970588, 18704696346822470087, 81978422471165944654
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2016

Keywords

Comments

Radius of convergence of g.f. A(x) is r = (5 - sqrt(17))/4 where r = r^2/(1-4*r+2*r^2) with A(r) = 1.
Compare g.f. with the identities:
(1) F(x)^2 = F( x^2/(1 - 4*x + 6*x^2) ) when F(x) = x/(1-2*x).
(2) C(x)^2 = C( x^2/(1 - 4*x + 4*x^2) ) when C(x) = (1-2*x - sqrt(1-4*x))/(2*x) is a g.f. of the Catalan numbers (A000108).
More generally, if
F(x)^2 = F( x^2/(1 - 2*a*x + 2*(a^2 - b)*x^2) ),
then
F( x/(1 + a*x + b*x^2) )^2 = F( x^2/(1 + a^2*x^2 + b^2*x^4) ).

Examples

			G.f.: A(x) = x + 2*x^2 + 6*x^3 + 20*x^4 + 71*x^5 + 262*x^6 + 994*x^7 + 3852*x^8 + 15183*x^9 + 60686*x^10 + 245410*x^11 + 1002300*x^12 +...
such that A( x^2/(1-4*x+2*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 64*x^5 + 258*x^6 + 1048*x^7 + 4288*x^8 + 17664*x^9 + 73223*x^10 + 305292*x^11 + 1279632*x^12  + 5389632*x^13 + 22800926*x^14 +...
The g.f. of A260650, F(x), begins:
A( x/(1 - 2*x) ) = x + 4*x^2 + 18*x^3 + 88*x^4 + 455*x^5 + 2444*x^6 + 13486*x^7 + 75912*x^8 + 433935*x^9 + 2511388*x^10 +...
and satisfies: F(x)^2 = F( x^2/(1 - 4*x)^2 ).
The series reversion of the g.f. A(x) begins:
Series_Reversion(A(x)) = x - 2*x^2 + 2*x^3 - 3*x^5 + 4*x^6 - 2*x^7 + 2*x^9 - 10*x^10 + 18*x^11 - 39*x^13 + 28*x^14 + 40*x^15 - 142*x^17 - 84*x^18 + 620*x^19 - 1735*x^21 + 260*x^22 + 4532*x^23 +...
which is related to A107087 by:
x/Series_Reversion(A(x)) = 1 + 2*x + 2*x^2 - x^4 + 2*x^6 - 5*x^8 + 12*x^10 - 30*x^12 + 82*x^14 - 233*x^16 + 668*x^18 - 1949*x^20 +...+ A107087(n)*x^(2*n) +...
The g.f. G(x) of A107087 begins:
G(x) = 1 + 2*x - x^2 + 2*x^3 - 5*x^4 + 12*x^5 - 30*x^6 + 82*x^7 - 233*x^8 + 668*x^9 - 1949*x^10 + 5802*x^11 - 17503*x^12 +...
where G(x)^2 = G(x^2) + 4*x.
Also, we have A(x/(1 + 2*x + 3*x^2))^2 = A(x^2/(1 + 4*x^2 + 9*x^4)), where the series begin:
A(x/(1 + 2*x + 3*x^2)) = x - x^3 - 2*x^5 + 6*x^7 - x^9 - 3*x^11 - 30*x^13 - 66*x^15 + 715*x^17 - 747*x^19 - 4028*x^21 + 9424*x^23 + 8790*x^25 +...
A(x^2/(1 + 4*x^2 + 9*x^4)) = x^2 - 2*x^4 - 3*x^6 + 16*x^8 - 10*x^10 - 28*x^12 - 14*x^14 - 72*x^16 + 1647*x^18 - 3014*x^20 - 10145*x^22 + 38784*x^24 +...
which is equal to A(x/(1 + 2*x + 3*x^2))^2.

Links

Crossrefs

Cf. A107087, A260650, A264224, A274483, A274478, A274479.
Cf. A264225, A357547, A357548.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-4*x+2*x^2 +x*O(x^n)) ) ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = -A( -x/(1 - 4*x) ). - Paul D. Hanna, Nov 30 2022
(2) A(x)^2 = A( x^2/(1 - 4*x + 2*x^2) ).
(3) A( x/(1 + 2*x + 3*x^2) )^2 = A( x^2/(1 + 4*x^2 + 9*x^4) ).
(4) A( x/(1 + 2*x) )^2 = x * A( x/(1 - 2*x) ).
(5) A( x/(1 - 2*x) )^2 = A( x^2/(1 - 8*x + 14*x^2) ).
Let G(x) denote the g.f. of A107087, where G(x)^2 = G(x^2) + 4*x, then g.f. A(x) satisfies:
(6) A(x) = x/(1-2*x) * G( A(x)^2 ),
(7) A(x) = Series_Reversion( x/(G(x)^2 - 2*x) ),
(8) G(x) = sqrt( x/Series_Reversion(A(x)) + 2*x ),
(9) G(x^2) = x/Series_Reversion(A(x)) - 2*x,
(10) A( x/(G(x)^2 - 2*x) ) = x,
(11) A( x/(G(x^2) + 2*x) ) = x,
(12) A(x)^2/(G(A(x)^4) + 2*A(x)^2) = x^2/(1 - 4*x + 2*x^2).

A274479 G.f. satisfies: A(x)^2 = A( x^2/(1 - 2*x - 4*x^2) ).

Original entry on oeis.org

1, 1, 4, 10, 34, 106, 361, 1219, 4252, 14932, 53263, 191533, 695233, 2540617, 9344050, 34546672, 128330533, 478653973, 1791816967, 6729202603, 25344884479, 95707901503, 362269464487, 1374203633335, 5223097370170, 19888174932226, 75856437036451, 289780169876749, 1108607284380835, 4246966803249139, 16290547536335716, 62562701811659506, 240540845892246253, 925825162823212429, 3567069859670052457, 13756707569545384033
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2016

Keywords

Comments

Compare g.f. with the identities:
(1) F(x)^2 = F( x^2/(1 - 2*x + 2*x^2) ) when F(x) = x/(1-x).
(2) M(x)^2 = M( x^2/(1 - 2*x) ) when M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).
a(n) = 1 (mod 3) for n>=1 (conjecture).
Radius of convergence of g.f. A(x) is r = 1/4 where r = r^2/(1-2*r-4*r^2) with A(1/4) = 1.
What is the limit a(n)/A000108(n) ? Note that A000108(n) = binomial(2*n,n)/(n+1) is the n-th Catalan number.

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 10*x^4 + 34*x^5 + 106*x^6 + 361*x^7 + 1219*x^8 + 4252*x^9 + 14932*x^10 + 53263*x^11 + 191533*x^12 +...
such that A( x^2/(1-2*x-4*x^2) ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 28*x^5 + 104*x^6 + 360*x^7 + 1306*x^8 + 4688*x^9 + 17106*x^10 + 62548*x^11 + 230570*x^12 + 853512*x^13 + 3176161*x^14 + 11866142*x^15 +...
The series reversion of the g.f. A(x) begins:
Series_Reversion(A(x)) = x - x^2 - 2*x^3 + 5*x^4 + 4*x^5 - 22*x^6 - 5*x^7 + 95*x^8 - 17*x^9 - 412*x^10 + 220*x^11 + 1790*x^12 - 1559*x^13 - 7771*x^14 +...
which is related to A264412 by:
x/Series_Reversion(A(x)) = 1 + x + 3*x^2 - 3*x^4 + 9*x^6 - 33*x^8 + 126*x^10 - 513*x^12 + 2214*x^14 - 9876*x^16 + 45045*x^18 - 209493*x^20 +...+ A264412(n)*x^(2*n) +...
The g.f. G(x) of A264412 begins:
G(x) = 1 + 3*x - 3*x^2 + 9*x^3 - 33*x^4 + 126*x^5 - 513*x^6 + 2214*x^7 - 9876*x^8 + 45045*x^9 - 209493*x^10 +...
where G(x)^2 = G(x^2) + 6*x.
Also, we have A(x/(1 + x + 3*x^2))^2 = A(x^2/(1 + x^2 + 9*x^4)), where the series begin:
A(x/(1 + x + 3*x^2)) = x - 3*x^5 + 3*x^9 + 81*x^13 - 840*x^17 + 3960*x^21 + 711*x^25 - 152145*x^29 + 1009254*x^33 - 1772820*x^37 + 1991277*x^41 +...
A(x^2/(1 + x^2 + 9*x^4)) = x^2 - 6*x^6 + 15*x^10 + 144*x^14 - 2157*x^18 + 13446*x^22 - 20817*x^26 - 420876*x^30 + 4282764*x^34 - 17051652*x^38 +...
which is equal to A(x/(1 + x + 3*x^2))^2.

Links

Crossrefs

Cf. A264412, A274478, A274484.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-2*x-4*x^2 +x*O(x^n)) ) ) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) satisfies: A( x/(1 + x + 3*x^2) )^2 = A( x^2/(1 + x^2 + 9*x^4) ).
Let G(x) denote the g.f. of A264412, where G(x)^2 = G(x^2) + 6*x, then g.f. A(x) satisfies:
(1) A(x) = x/(1-x) * G( A(x)^2 ),
(2) G(x^2) = x/Series_Reversion(A(x)) - x,
(3) A( x/(G(x^2) + x) ) = x,
(4) A(x)^2/(G(A(x)^4) + A(x)^2) = x^2/(1 - 2*x - 4*x^2).
Showing 1-2 of 2 results.

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