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.

A351771 Given g.f. A(x), the even bisections of both A(x) and A(x)^2 are equal, and the odd bisections of both A(x)^2 and A(x)^3 are equal (after the initial terms).

Original entry on oeis.org

1, 1, -1, 3, -7, 28, -79, 350, -1075, 5020, -16180, 78023, -259417, 1278340, -4343642, 21740636, -75065787, 380161308, -1328887420, 6792111260, -23975385148, 123448657904, -439228736887, 2275311657814, -8148868193557, 42427160829508, -152792221834364
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2022

Keywords

Comments

a(2*n+1) = A352383(n) for n >= 0.

Examples

			G.f. A(x) = 1 + x - 1*x^2 + 3*x^3 - 7*x^4 + 28*x^5 - 79*x^6 + 350*x^7 - 1075*x^8 + 5020*x^9 - 16180*x^10 + 78023*x^11 + ...
Compare A(x) with the coefficients in the following series expansions:
A(x)^2 = 1 + 2*x - 1*x^2 + 4*x^3 - 7*x^4 + 36*x^5 - 79*x^6 + 444*x^7 - 1075*x^8 + 6324*x^9 - 16180*x^10 + 97872*x^11 + ...
A(x)^3 = 1 + 3*x + 0*x^2 + 4*x^3 - 3*x^4 + 36*x^5 - 40*x^6 + 444*x^7 - 579*x^8 + 6324*x^9 - 9000*x^10 + 97872*x^11 + ...
A(x)^4 = 1 + 4*x + 2*x^2 + 4*x^3 + 3*x^4 + 36*x^5 + 16*x^6 + 444*x^7 + 121*x^8 + 6324*x^9 + 1040*x^10 + 97872*x^11 + ...
which illustrate the properties that the coefficients of x^k for even k in A(x) and A(x)^2 are equal, and that the coefficients of x^k for odd k > 1 in A(x)^2, A(x)^3, and A(x)^4 are equal.
Related series.
(1) Notice that A(x)^2 - A(x) forms an odd function:
A(x)^2 - A(x) = x + x^3 + 8*x^5 + 94*x^7 + 1304*x^9 + 19849*x^11 + 320600*x^13 + 5396108*x^15 + ...
such that the series reversion begins
Series_Reversion( A(x)^2 - A(x) ) = x - x^3 - 5*x^5 - 42*x^7 - 429*x^9 - 4862*x^11 - 58786*x^13 - ...
which equals x - x*(C(x) + C(-x))/2, where C(x) = x + C(x)^2:
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ...
and is the Catalan power series C(x) = (1 - sqrt(1-4*x))/2.
(2) Also, the coefficients in the following series form a bisection of A(x):
(A(x)^4 - A(x)^3 - x)/2 = x^2 + 3*x^4 + 28*x^6 + 350*x^8 + 5020*x^10 + 78023*x^12 + 1278340*x^14 + ... + A352383(n)*x^(2*n+2) + ...
(3) Further, a series bisection of A(x)^2, A(x)^3, and A(x)^4 is
(A(x) - A(-x))^3/2 = 4*x^3 + 36*x^5 + 444*x^7 + 6324*x^9 + 97872*x^11 + 1598940*x^13 + 27136744*x^15 + ... + 4*A352384(n)*x^(2*n+3) + ...
which is equal to 4*x*F(x^2), where F( x*(1+x)^3/(1+2*x)^6 ) = x, and
F(x) = x + 9*x^2 + 111*x^3 + 1581*x^4 + 24468*x^5 + 399735*x^6 + 6784186*x^7 + ... + A352384(n)*x^(n+1) + ...
with
(F(x)/x)^(1/3) = 1 + 3*x + 28*x^2 + 350*x^3 + 5020*x^4 + 78023*x^5 + 1278340*x^6 + ... + A352383(n)*x^n + ...
(4) The above observations lead to the composition of functions
Series_Reversion(A(x) - 1) = [x - x*(C(x) + C(-x))/2] o (x + x^2)
which is equivalent to
Series_Reversion(A(x) - 1) = x*(1+x)*(3 + 2*x + sqrt(1-4*x-4*x^2))/4.

Crossrefs

Cf. A352383, A352384, A000108.

Programs

  • Mathematica
    CoefficientList[1 + InverseSeries[Series[x*(1 + x)*(3 + 2*x + Sqrt[1 - 4*x - 4*x^2])/4, {x, 0, 30}], x], x] (* Vaclav Kotesovec, Mar 15 2022 *)
  • PARI
    /* Using Series Reversion */
{a(n) = my(A = 1 + serreverse( x*(1+x)*(3 + 2*x + sqrt(1-4*x-4*x^2 +x^2*O(x^n)))/4)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* From [x^(2*n)] A(x) - A(x)^2 = 0 and [x^(2*n+1)] A(x)^2 - A(x)^3 = 0  */
{a(n) = my(A = 1 + x +x^2*O(x^n));
for(k=2,n, if(k%2==0,
A = A + x^k*polcoeff(A^1 - A^2,k),
A = A + x^k*polcoeff(A^2 - A^3,k)));
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1a) [x^(2*n)] A(x) = [x^(2*n)] A(x)^2 for n >= 1.
(1b) [x^(2*n+1)] A(x)^2 = [x^(2*n+1)] A(x)^3 for n >= 1.
(1c) [x^(2*n+1)] A(x)^3 = [x^(2*n+1)] A(x)^4 for n >= 1.
(2) (A(x) - A(-x))/2 = x/(A(x)*A(-x)).
(3a) (A(x)^2 + A(-x)^2)/2 = (A(x) + A(-x))/2.
(3b) (A(x)^2 - A(-x)^2)/2 = 2*x + (A(x) - A(-x))^3/2.
(4) A(x)^2 = 2*x + (A(x) + A(-x))/2 + (A(x) - A(-x))^3/2.
(5a) A(x)^2 = 2*x + (A(x)^2 + A(-x)^2)/2 + (A(x) - A(-x))^3/2.
(5b) A(x)^3 = 3*x + (A(x)^3 + A(-x)^3)/2 + (A(x) - A(-x))^3/2.
(5c) A(x)^4 = 4*x + (A(x)^4 + A(-x)^4)/2 + (A(x) - A(-x))^3/2.
(6) A(x)^4 - A(x)^3 = x + x*(A(x) - A(-x)).
(7) A(-x) = (A(x)^2 + sqrt(A(x)^4 - 8*x*A(x)))/(2*A(x)).
(8) (A(x) - A(-x))^3/2 = 4*x*F(x^2), where F(x) = Series_Reversion( x*(1+x)^3/(1+2*x)^6 ).
(9) A(x)^2 - A(x) = Series_Reversion( x - x*(C(x) + C(-x))/2 ), where C(x) = x + C(x)^2 is the Catalan power series (A000108).
(10) A(x) = 1 + Series_Reversion( x*(1+x)*(3 + 2*x + sqrt(1-4*x-4*x^2))/4 ).
(11) 0 = 2*x^2 + A(x)*(1 - A(x))*(1 + 2*A(x))*x + A(x)^4*(1 - A(x))^2.

A352384 G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x))^6 / (1 + x*A(x))^3.

Original entry on oeis.org

1, 9, 111, 1581, 24468, 399735, 6784186, 118444293, 2113587804, 38377421060, 706774205943, 13170180868299, 247862354439196, 4704490506021162, 89949748461476772, 1730889637195688117, 33495746280466024908
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2022

Keywords

Comments

Self-convolution cube root yields A352383.

Examples

			G.f.: A(x) = 1 + 9*x + 111*x^2 + 1581*x^3 + 24468*x^4 + 399735*x^5 + 6784186*x^6 + 118444293*x^7 + 2113587804*x^8 + 38377421060*x^9 + ...
where
A(x)^(1/3) = (1 + 2*x*A(x))^2/(1 + x*A(x)) = 1 + 3*x + 28*x^2 + 350*x^3 + 5020*x^4 + 78023*x^5 + 1278340*x^6 + ... + A352383(n)*x^n + ...

Crossrefs

Cf. A352383, A351771.

Programs

  • Mathematica
    CoefficientList[(InverseSeries[Series[x*(1 + x)^3/(1 + 2*x)^6, {x, 0, 20}], x]/x), x] (* Vaclav Kotesovec, Mar 15 2022 *)
  • PARI
    /* Using Series Reversion */
{a(n) = my(A = (1/x)*serreverse( x*(1+x)^3/(1+2*x +x^2*O(x^n))^6 )); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = (1 + 2*x*A(x))^6 / (1 + x*A(x))^3.
(2) A( x*(1+x)^3/(1+2*x)^6 ) = (1+2*x)^6/(1+x)^3.
(3) A(x) = (1/x) * Series_Reversion( x*(1+x)^3/(1+2*x)^6 ).
(4) x = ( A(x)^(1/3) - 4 + sqrt( A(x)^(2/3) + 8*A(x)^(1/3) ) ) / (8*A(x)).
(5) 0 = 4*A(x)^2*x^2 + (4 - A(x)^(1/3))*A(x)*x + (1 - A(x)^(1/3)).
a(n) ~ sqrt(278513 - 1003421/sqrt(13)) * 2^(3*n + 3/2) * (587 - 143*sqrt(13))^n / (sqrt(Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Mar 15 2022
D-finite with recurrence 3*n *(3*n+2) *(2*n+3) *(3*n+1) *(5114882323*n -3577270936)*(n+1)*a(n) -8*n*(12009974028164*n^5 +15575274162434*n^4 +20452834455*n^3 -7391009529770*n^2 -2779978786544*n -151626455514) *a(n-1) +64*(29840008960856*n^6 -7909817331616*n^5 -21378617546230*n^4 -22395081360175*n^3 +39992783684339*n^2 -16585158398179*n +2497181632755) *a(n-2) -4608*(6*n-11) *(6*n-7) *(3*n-4) *(2*n-3) *(3*n-5) *(4265440*n -810084569)*a(n-3)=0. - R. J. Mathar, Jul 20 2023

A352701 G.f. (1/x)*Series_Reversion( x*(1-x)*(3 - 2*x + sqrt(1+4*x-4*x^2))/4 ).

Original entry on oeis.org

1, 1, 3, 7, 28, 79, 350, 1075, 5020, 16180, 78023, 259417, 1278340, 4343642, 21740636, 75065787, 380161308, 1328887420, 6792111260, 23975385148, 123448657904, 439228736887, 2275311657814, 8148868193557, 42427160829508
Offset: 0

Views

Author

Paul D. Hanna, Mar 29 2022

Keywords

Comments

Essentially an unsigned version of A351771 (after dropping the initial term).
a(2*n) = A352383(n) for n >= 0.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 28*x^4 + 79*x^5 + 350*x^6 + 1075*x^7 + 5020*x^8 + 16180*x^9 + 78023*x^10 + 259417*x^11 + ...
such that A(x) = (1/x)*Series_Reversion(x*G(x)) and A(x*G(x)) = 1/G(x),
where G(x) = (1-x)*(3 - 2*x + sqrt(1+4*x-4*x^2))/4, which starts
G(x) = 1 - x - x^2 + 3*x^3 - 8*x^4 + 26*x^5 - 92*x^6 + 344*x^7 - 1336*x^8 + 5336*x^9 - 21776*x^10 + ...
Let B(x) =  Series_Reversion( x*(1-x^2)/(1+x^2)^3 ),
B(x) = x + 4*x^3 + 39*x^5 + 496*x^7 + 7180*x^9 + 112236*x^11 + 1846082*x^13 + 31485120*x^15 + ...,
then A(x) = 1 + x*A(x)^2 + B(x)^2, where
B(x)^2 = x^2 + 8*x^4 + 94*x^6 + 1304*x^8 + 19849*x^10 + 320600*x^12 + 5396108*x^14 + 93615864*x^16 + ...

Crossrefs

Cf. A351771, A352383.

Programs

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

Formula

The g.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x*(1-x)*(3 - 2*x + sqrt(1+4*x-4*x^2))/4 );
(2) (A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2;
(3) ((A(x) + A(-x))/2)^3 = F(x^2), where F(x) = (1/x)*Series_Reversion( x*(1+x)^3/(1+2*x)^6 );
(4) 1 - x*(A(x) - A(-x))/2 = x/Series_Reversion( x - x*(C(x) + C(-x))/2 ), where C(x) = (1 - sqrt(1-4*x))/2 is the Catalan function (A000108);
(5a) (1/A(x) + 1/A(-x))/2 = ( 1 - x*(A(x) - A(-x))/2 )^2;
(5b) (1/A(x) - 1/A(-x))/2 = (-x)/(1 - 2*x*(A(x) - A(-x))/2);
(6a) (1/A(x)^2 + 1/A(-x)^2)/2 = ( 1 - x*(A(x) - A(-x))/2 )^3.
(6b) (1/A(x)^2 - 1/A(-x)^2)/2 = -2*x*(1 - x*(A(x) - A(-x))/2)^2/( 1 - x*(A(x) - A(-x)) ).
Let B(x) = Series_Reversion( x*(1-x^2)/(1+x^2)^3 ), then
(7) A(x) = (1 - sqrt(1 - 4*x - 4*x*B(x)^2))/(2*x);
(8) A(x) - x*A(x)^2 = A(-x) + x*A(-x)^2 = 1 + B(x)^2;
(9) 1 - x*(A(x) - A(-x))/2 = 1/(1 + B(x)^2);
(10) 1/A(x) = 1/(1 + B(x)^2)^2 - x*(1 + B(x)^2)/(1 - B(x)^2);
(10a) (1/A(x) + 1/A(-x))/2 = 1/(1 + B(x)^2)^2;
(10b) (1/A(x) - 1/A(-x))/2 = (-x)*(1 + B(x)^2)/(1 - B(x)^2);
(11) 1/A(x)^2 = 1/(1 + B(x)^2)^3 - 2*x/(1 - B(x)^4);
(11a) (1/A(x)^2 + 1/A(-x)^2)/2 = 1/(1 + B(x)^2)^3;
(11b) (1/A(x)^2 - 1/A(-x)^2)/2 = -2*x/(1 - B(x)^4).
Showing 1-3 of 3 results.

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