A363556 Expansion of g.f. A(x) satisfying 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n^2) / Product_{k=2..n+2} (1 - x^k*A(x)^(2*k+1)).

1, 1, 4, 20, 116, 729, 4835, 33308, 236040, 1709710, 12601576, 94209348, 712665643, 5445033052, 41957720686, 325702058100, 2544612673624, 19993230400674, 157880283331190, 1252345341860312, 9974116387535508, 79727867712667593, 639419309923767255, 5143690907082615272

Offset: 0

Author

Paul D. Hanna, Jul 09 2023

Keywords

• nonn

Comments

Related identities:

(1) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * D(x)^(n^2) / Product_{k=1..n+1} (1 - x^k*D(x)^(2*k+1)), where D(x) = 1 + x*D(x)^3 is the g.f. of A001764.

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

(3) Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * y^(n^2) / Product_{k=2..n+2} (1 - x^k*y^(2*k+1)) = (1 - y) + y^4*x + (y^5 - y^6)*x^2 - y^8*x^3 + y^10*x^4.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 116*x^4 + 729*x^5 + 4835*x^6 + 33308*x^7 + 236040*x^8 + 1709710*x^9 + 12601576*x^10 + ...
such that A = A(x) satisfies
0 = 1/(1-x^2*A^5) - A/((1-x^2*A^5)*(1-x^3*A^7)) + x*A^4/((1-x^2*A^5)*(1-x^3*A^7)*(1-x^4*A^9)) - x^3*A^9/((1-x^2*A^5)*(1-x^3*A^7)*(1-x^4*A^9)*(1-x^5*A^11)) + x^6*A^16/((1-x^2*A^5)*(1-x^3*A^7)*(1-x^4*A^9)*(1-x^5*A^11)*(1-x^6*A^13)) -+ ...
Compare to an identity of the function D = 1 + x*D^3 (A001764), where
0 = 1/(1-x*D^3) - D/((1-x*D^3)*(1-x^2*D^5)) + x*D^4/((1-x*D^3)*(1-x^2*D^5)*(1-x^3*D^7)) - x^3*D^9/((1-x*D^3)*(1-x^2*D^5)*(1-x^3*D^7)*(1-x^4*D^9)) + x^6*D^16/((1-x*D^3)*(1-x^2*D^5)*(1-x^3*D^7)*(1-x^4*D^9)*(1-x^5*D^11)) -+ ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A082395, A363555.

Programs

PARI

/* As a solution to a polynomial equation */
{a(n) = my(A=1); for(i=1, n, A = 1 + A^4*x + (A^5 - A^6)*x^2 - A^8*x^3 + A^10*x^4 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

/* By Series Reversion involving a g.f. for A082395 */
{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 = sqrt( (1/x)*serreverse( x/G082395^2 ) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

/* By Definition */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=0,2*sqrtint(#A), (-1)^m * (x)^(m*(m-1)/2) * Ser(A)^(m^2) / prod(k=2,m+2, (1 - 1*x^k*Ser(A)^(2*k+1) ) )),#A-1);); H=A; A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

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

(1) A(x) = 1 + A(x)^4*x + (A(x)^5 - A(x)^6)*x^2 - A(x)^8*x^3 + A(x)^10*x^4.

(2) A(x) = F(x*A(x)) where F(x) = A(x/F(x)) is the g.f. of A363555.

(3) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) = (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).

(4) 0 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * A(x)^(n^2) / Product_{k=2..n+2} (1 - x^k*A(x)^(2*k+1)).