A363555 - OEIS

%I #30 Jul 20 2023 10:06:28

%S 1,1,3,10,39,163,719,3291,15495,74568,365188,1814220,9120733,46316195,

%T 237229711,1224150186,6358008567,33211790566,174368764180,

%U 919634519638,4870040209108,25885098736539,138044748197031,738439479552465,3961167293262753,21303283844252223

%N Expansion of g.f. A(x) satisfying 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2) / Product_{k=2..n+2} (1 - x^k*A(x)^(k+1)).

%C Related identities:

%C (1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * C(x)^(n*(n+1)/2) / Product_{k=1..n+1} (1 - x^k*C(x)^(k+1)), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

%C (2) 1 = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * F(x)^(n*(n+1)/2) / Product_{k=1..n+1} (1 - x^k*F(x)^k), which holds as a formal power series for all x and F(x).

%C (3) Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * y^(n*(n+1)/2) / Product_{k=2..n+2} (1 - x^k*y^(k+1)) = 1-y + y^3*x + (y^3 - y^4)*x^2 - y^5*x^3 + y^6*x^4.

%H Paul D. Hanna, <a href="/A363555/b363555.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

%F (1) A(x) = 1 + A(x)^3*x + (A(x)^3 - A(x)^4)*x^2 - A(x)^5*x^3 + A(x)^6*x^4.

%F (2) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = (1 + x - sqrt(1 - 2*x - 3*x^2))/(2*x*(1-x^2)) is the g.f. of A082395 (with an offset of 0).

%F (3) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2) / Product_{k=2..n+2} (1 - x^k*A(x)^(k+1)).

%e G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 39*x^4 + 163*x^5 + 719*x^6 + 3291*x^7 + 15495*x^8 + 74568*x^9 + 365188*x^10 + ...

%e such that A = A(x) satisfies

%e 0 = 1/(1-x^2*A^3) - A/((1-x^2*A^3)*(1-x^3*A^4)) + x*A^3/((1-x^2*A^3)*(1-x^3*A^4)*(1-x^4*A^5)) - x^3*A^6/((1-x^2*A^3)*(1-x^3*A^4)*(1-x^4*A^5)*(1-x^5*A^6)) + x^6*A^10/((1-x^2*A^3)*(1-x^3*A^4)*(1-x^4*A^5)*(1-x^5*A^6)*(1-x^6*A^7)) -+ ...

%e Compare to an identity of the Catalan function C = 1 + x*C^2, where

%e 0 = 1/(1-x*C^2) - C/((1-x*C^2)*(1-x^2*C^3)) + x*C^3/((1-x*C^2)*(1-x^2*C^3)*(1-x^3*C^4)) - x^3*C^6/((1-x*C^2)*(1-x^2*C^3)*(1-x^3*C^4)*(1-x^4*C^5)) + x^6*C^10/((1-x*C^2)*(1-x^2*C^3)*(1-x^3*C^4)*(1-x^4*C^5)*(1-x^5*C^6)) -+ ...

%o (PARI) /* As a solution to a polynomial equation */

%o {a(n) = my(A=1); for(i=1,n, A = 1 + A^3*x + (A^3 - A^4)*x^2 - A^5*x^3 + A^6*x^4 +x*O(x^n)); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) /* By Series Reversion involving a g.f. of A082395 */

%o {a(n) = my(A=1, G082395 = (1 + x - sqrt(1 - 2*x - 3*x^2 +O(x^(n+2))))/(2*x*(1-x^2)) ); A = (1/x)*serreverse( x/G082395 ); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) /* By Definition */

%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = polcoeff( sum(m=0,2*sqrtint(#A), (-1)^m * (x)^(m*(m-1)/2) * Ser(A)^(m*(m+1)/2) / prod(k=2,m+2, (1 - x^k*Ser(A)^(k+1) ) )),#A-1);); A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A082395, A363556.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 09 2023