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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.

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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 1-2 of 2 results.

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