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.

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A357547 a(n) = coefficient of x^n in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).

Original entry on oeis.org

1, 2, 9, 38, 176, 832, 4039, 19938, 99861, 506042, 2590099, 13370898, 69540016, 364028992, 1916585714, 10142059868, 53911982971, 287736310102, 1541243386819, 8282387269058, 44638363790176, 241216694913632, 1306608966475854, 7092980525443588, 38581011402034156
Offset: 1

Views

Author

Paul D. Hanna, Dec 01 2022

Keywords

Comments

Radius of convergence is r = (sqrt(41) - 5)/8, where r = r^2/(1 - 4*r - 4*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 = 6.

Examples

			G.f.: A(x) = x + 2*x^2 + 9*x^3 + 38*x^4 + 176*x^5 + 832*x^6 + 4039*x^7 + 19938*x^8 + 99861*x^9 + 506042*x^10 + 2590099*x^11 + 13370898*x^12 + ...
where A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 112*x^5 + 585*x^6 + 3052*x^7 + 16018*x^8 + 84384*x^9 + 446384*x^10 + 2370240*x^11 + 12631104*x^12 + ...
(x*A(x))^(1/2) = 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 + ... + A357785(n)*x^n + ...
x/Series_Reversion(A(x)) = 1 + 2*x + 5*x^2 - 10*x^4 + 50*x^6 - 305*x^8 + 2025*x^10 - 14400*x^12 + 107500*x^14 - 829415*x^16 + 6559700*x^18 - 52908950*x^20 + ...

Links

Crossrefs

Cf. A357785, A264224, A274483, A274484, A357548.

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)) ) )
); 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 + 6*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 6^2*x^4) ).
(2) A(x) = -A( -x/(1 - 4*x) ).
(3.a) A(x)^2 = A( x^2/(1 - 4*x - 4*x^2) ).
(3.b) A(x)^2 = -A( -x^2/(1 - 4*x - 8*x^2) ).
(4.a) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 8*x^2) ).
(4.b) A( x/(1 + 2*x) )^2 = -A( -x^2/(1 - 12*x^2) ).
(4.c) A( x/(1 + 2*x) )^2 = A( -x/(1 - 2*x) )^2.

A357786 a(n) = coefficient of x^n, n >= 1, in A(x) such that: A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ) * sqrt(1 - 4*x - 8*x^2).

Original entry on oeis.org

1, 1, 5, 20, 98, 483, 2499, 13182, 71030, 388484, 2152982, 12061840, 68212585, 388886050, 2232764700, 12898728750, 74923372563, 437303591874, 2563373794884, 15083551143318, 89060360731377, 527477003037984, 3132774700791126, 18652891302520806, 111314950683514698
Offset: 1

Views

Author

Paul D. Hanna, Dec 03 2022

Keywords

Comments

Self convolution equals A357548.
Radius of convergence is r = (sqrt(57) - 5)/16, where r = r^2/(1 - 4*r - 8*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 + 5*x^3 + 20*x^4 + 98*x^5 + 483*x^6 + 2499*x^7 + 13182*x^8 + 71030*x^9 + 388484*x^10 + 2152982*x^11 + ...
such that
A(x)^2 = A( x^2/(1 - 4*x - 8*x^2) ) * sqrt(1 - 4*x - 8*x^2)
where
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 50*x^5 + 261*x^6 + 1362*x^7 + 7344*x^8 + 40112*x^9 + 222338*x^10 + ... + A357548(n)*x^(n+1) + ...

Crossrefs

Cf. A357548, A357785.

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)) )*sqrt(1 - 4*x - 8*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 - 8*x^2) ) * sqrt(1 - 4*x - 8*x^2).
(3) A( x/(1 + 2*x) )^2 = A( x^2/(1 - 12*x^2) ) * sqrt(1 - 12*x^2) / (1 + 2*x).
(4) A( x/(1 + 2*x + 8*x^2) )^2 = A( x^2/(1 + 2^2*x^2 + 8^2*x^4) ) * sqrt(1 + 2^2*x^2 + 8^2*x^4) / (1 + 2*x + 8*x^2).
Showing 1-2 of 2 results.

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