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.

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).

A357548 a(n) = coefficient of x^n in A(x) where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).

Original entry on oeis.org

1, 2, 11, 50, 261, 1362, 7344, 40112, 222338, 1245476, 7043605, 40153390, 230518723, 1331576430, 7733934030, 45138530004, 264596552838, 1557101158092, 9195520745412, 54477134410680, 323668083179382, 1928047124332764, 11512382184408072, 68889282756213840
Offset: 1

Views

Author

Paul D. Hanna, Dec 01 2022

Keywords

Comments

Radius of convergence is r = (sqrt(57) - 5)/16, where r = r^2/(1 - 4*r - 8*r^2), with A(r) = 1.
Related 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) );
here, a = 2, b = 8.

Examples

			G.f.: A(x) = x + 2*x^2 + 11*x^3 + 50*x^4 + 261*x^5 + 1362*x^6 + 7344*x^7 + 40112*x^8 + 222338*x^9 + 1245476*x^10 + 7043605*x^11 + 40153390*x^12 + ...
where A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 26*x^4 + 144*x^5 + 843*x^6 + 4868*x^7 + 28378*x^8 + 165664*x^9 + 971013*x^10 + 5708132*x^11 + 33660362*x^12 + ...
(x*A(x))^(1/2) = x + x^2 + 5*x^3 + 20*x^4 + 98*x^5 + 483*x^6 + 2499*x^7 + 13182*x^8 + 71030*x^9 + 388484*x^10 + ... + A357786(n)*x^n + ...
x/Series_Reversion(A(x)) = 1 + 2*x + 7*x^2 - 21*x^4 + 147*x^6 - 1260*x^8 + 11907*x^10 - 120540*x^12 + 1279047*x^14 - 14029428*x^16 + 157788183*x^18 + ...

Links

Crossrefs

Cf. A357786, A264224, A274483, A274484, A357547.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, #binary(n+1),
A = sqrt( subst(A, x, x^2/(1 - 4*x - 8*x^2 +x*O(x^n)) ) )
); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x/(1 + 2*x + 8*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 8^2*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3) A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ).
(4) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 12*x^2) ).
(5) A( x/(1 + 4*x) )^2 = A( x^2/(1 + 4*x - 8*x^2) ).

A357785 a(n) = coefficient of x^n, n >= 1, in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ) * sqrt(1 - 4*x - 4*x^2).

Original entry on oeis.org

1, 1, 4, 15, 65, 291, 1356, 6474, 31555, 156315, 784924, 3986534, 20444676, 105728100, 550735400, 2886924190, 15217019595, 80600822575, 428766983300, 2289637381800, 12268642450420, 65941128441080, 355396218177760, 1920215555772550, 10398415258863275
Offset: 1

Views

Author

Paul D. Hanna, Dec 03 2022

Keywords

Comments

Self convolution equals A357547.
Radius of convergence is r = (sqrt(41) - 5)/8, where r = r^2/(1 - 4*r - 4*r^2), with A(r) = sqrt(r).
Related 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).

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 15*x^4 + 65*x^5 + 291*x^6 + 1356*x^7 + 6474*x^8 + 31555*x^9 + 156315*x^10 + 784924*x^11 + 3986534*x^12 + ...
such that
A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ) * sqrt(1 - 4*x - 4*x^2)
where
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 38*x^5 + 176*x^6 + 832*x^7 + 4039*x^8 + 19938*x^9 + 99861*x^10 + ... + A357547(n)*x^(n+1) + ...

Crossrefs

Cf. A357547, A357786.

Programs

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

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x) = -A( -x/(1 - 4*x) ) * sqrt(1 - 4*x).
(2) A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ) * sqrt(1 - 4*x - 4*x^2).
(3) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 8*x^2) ) * sqrt(1 - 8*x^2) / (1 + 2*x).
(4) A( x/(1 + 2*x + 6*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 6^2*x^4) ) * sqrt(1 + 2^2*x^2 + 6^2*x^4) / (1 + 2*x + 6*x^2).
Showing 1-3 of 3 results.

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