cp's OEIS Frontend

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).

%I A351771 #14 Mar 16 2022 16:47:02
%S A351771 1,1,-1,3,-7,28,-79,350,-1075,5020,-16180,78023,-259417,1278340,
%T A351771 -4343642,21740636,-75065787,380161308,-1328887420,6792111260,
%U A351771 -23975385148,123448657904,-439228736887,2275311657814,-8148868193557,42427160829508,-152792221834364
%N 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).
%C A351771 a(2*n+1) = A352383(n) for n >= 0.
%F A351771 G.f. A(x) satisfies:
%F A351771 (1a) [x^(2*n)] A(x) = [x^(2*n)] A(x)^2 for n >= 1.
%F A351771 (1b) [x^(2*n+1)] A(x)^2 = [x^(2*n+1)] A(x)^3 for n >= 1.
%F A351771 (1c) [x^(2*n+1)] A(x)^3 = [x^(2*n+1)] A(x)^4 for n >= 1.
%F A351771 (2) (A(x) - A(-x))/2 = x/(A(x)*A(-x)).
%F A351771 (3a) (A(x)^2 + A(-x)^2)/2 = (A(x) + A(-x))/2.
%F A351771 (3b) (A(x)^2 - A(-x)^2)/2 = 2*x + (A(x) - A(-x))^3/2.
%F A351771 (4) A(x)^2 = 2*x + (A(x) + A(-x))/2 + (A(x) - A(-x))^3/2.
%F A351771 (5a) A(x)^2 = 2*x + (A(x)^2 + A(-x)^2)/2 + (A(x) - A(-x))^3/2.
%F A351771 (5b) A(x)^3 = 3*x + (A(x)^3 + A(-x)^3)/2 + (A(x) - A(-x))^3/2.
%F A351771 (5c) A(x)^4 = 4*x + (A(x)^4 + A(-x)^4)/2 + (A(x) - A(-x))^3/2.
%F A351771 (6) A(x)^4 - A(x)^3 = x + x*(A(x) - A(-x)).
%F A351771 (7) A(-x) = (A(x)^2 + sqrt(A(x)^4 - 8*x*A(x)))/(2*A(x)).
%F A351771 (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 ).
%F A351771 (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).
%F A351771 (10) A(x) = 1 + Series_Reversion( x*(1+x)*(3 + 2*x + sqrt(1-4*x-4*x^2))/4 ).
%F A351771 (11) 0 = 2*x^2 + A(x)*(1 - A(x))*(1 + 2*A(x))*x + A(x)^4*(1 - A(x))^2.
%e A351771 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 + ...
%e A351771 Compare A(x) with the coefficients in the following series expansions:
%e A351771 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 + ...
%e A351771 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 + ...
%e A351771 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 + ...
%e A351771 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.
%e A351771 Related series.
%e A351771 (1) Notice that A(x)^2 - A(x) forms an odd function:
%e A351771 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 + ...
%e A351771 such that the series reversion begins
%e A351771 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 - ...
%e A351771 which equals x - x*(C(x) + C(-x))/2, where C(x) = x + C(x)^2:
%e A351771 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 + ...
%e A351771 and is the Catalan power series C(x) = (1 - sqrt(1-4*x))/2.
%e A351771 (2) Also, the coefficients in the following series form a bisection of A(x):
%e A351771 (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) + ...
%e A351771 (3) Further, a series bisection of A(x)^2, A(x)^3, and A(x)^4 is
%e A351771 (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) + ...
%e A351771 which is equal to 4*x*F(x^2), where F( x*(1+x)^3/(1+2*x)^6 ) = x, and
%e A351771 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) + ...
%e A351771 with
%e A351771 (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 + ...
%e A351771 (4) The above observations lead to the composition of functions
%e A351771 Series_Reversion(A(x) - 1) = [x - x*(C(x) + C(-x))/2] o (x + x^2)
%e A351771 which is equivalent to
%e A351771 Series_Reversion(A(x) - 1) = x*(1+x)*(3 + 2*x + sqrt(1-4*x-4*x^2))/4.
%t A351771 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 *)
%o A351771 (PARI) /* Using Series Reversion */
%o A351771 {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)}
%o A351771 for(n=0,30,print1(a(n),", "))
%o A351771 (PARI) /* From [x^(2*n)] A(x) - A(x)^2 = 0 and [x^(2*n+1)] A(x)^2 - A(x)^3 = 0  */
%o A351771 {a(n) = my(A = 1 + x +x^2*O(x^n));
%o A351771 for(k=2,n, if(k%2==0,
%o A351771 A = A + x^k*polcoeff(A^1 - A^2,k),
%o A351771 A = A + x^k*polcoeff(A^2 - A^3,k)));
%o A351771 polcoeff(A,n)}
%o A351771 for(n=0,30,print1(a(n),", "))
%Y A351771 Cf. A352383, A352384, A000108.
%K A351771 sign
%O A351771 0,4
%A A351771 _Paul D. Hanna_, Mar 14 2022