A268300 - OEIS

%I #41 Mar 11 2024 23:10:47

%S 1,7,119,2118,42523,914922,20745494,487390092,11764545555,

%T 289962708802,7267069560834,184626340341588,4744080078088734,

%U 123075608359376932,3219261610951795084,84806249132678044440,2248017950109054256899,59917503707743905031346,1604813748929693765997450,43170742498490205711682564,1165893490887496323343495146,31598783791475055433157814444,859179326846115018832395000820

%N G.f. satisfies: -1 = Product_{n>=1} (1-x^n) * (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)), where g.f. A(x) = Sum_{n>=0} a(n)*2*(x/4)^n.

%C The g.f. utilizes the Jacobi Triple Product: Product_{n>=1} (1-x^n)*(1 - x^n/a)*(1 - x^(n-1)*a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.

%H Paul D. Hanna, <a href="/A268300/b268300.txt">Table of n, a(n) for n = 0..300</a>

%F Given g.f. A(x) = Sum_{n>=0} a(n) * 2*(x/4)^n, then g.f. also satisfies:

%F (1) -1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n,

%F (2) A(x) = 1 / Product_{n>=1} (1-x^n) * (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)),

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

%F (4) x = Sum_{n>=1} A268299(n) * x^n / A(x)^n.

%F a(n) is odd iff n = 2^k for k>=0 or n=0 (conjecture).

%F a(n) ~ c * d^n / n^(3/2), where d = 29.10159109069361717048796233905065832... and c = 0.57417747020768285925989822148605305... . - _Vaclav Kotesovec_, Mar 02 2016

%F Formula (2) can be rewritten as the functional equation y = 1 / (QPochhammer(x) * QPochhammer(y,x) / (1-y) * QPochhammer(1/(x*y),x) / (1 - 1/(x*y))). - _Vaclav Kotesovec_, Jan 19 2024

%e G.f.: A(x) = 2 + 7*2*x/4 + 119*2*x^2/4^2 + 2118*2*x^3/4^3 + 42523*2*x^4/4^4 + 914922*2*x^5/4^5 + 20745494*2*x^6/4^6 + 487390092*2*x^7/4^7 + 11764545555*2*x^8/4^8 + 289962708802*2*x^9/4^9 + 7267069560834*2*x^10/4^10 +...

%e where g.f. A(x) satisfies the Jacobi Triple Product:

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

%e also

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

%e RELATED SERIES.

%e 1/A(x) = 1/2 - 7/2*x/4 - 70/2*x^2/4^2 - 795/2*x^3/4^3 - 13802/2*x^4/4^4 - 277782/2*x^5/4^5 - 6093708/2*x^6/4^6 - 139376659/2*x^7/4^7 - 3297234754/2*x^8/4^8 - 79988099074/2*x^9/4^9 +...+ A268301(n)/2*x^n/4^n +...

%e Series_Reversion( x/A(x) ) = 2*x + 7*x^2 + 84*x^3 + 1240*x^4 + 20942*x^5 + 382344*x^6 + 7354688*x^7 + 146810440*x^8 + 3012778758*x^9 + 63167322872*x^10 +...+ A268299(n)*x^n +..., an integer series.

%e Let J(x) = Sum_{n>=1} x^(n*(n-1)/2) * (A(x)^n + 1/A(x)^(n-1)),

%e then J(x) is an integer series:

%e J(x) = 3 + 8*x + 28*x^2 + 144*x^3 + 736*x^4 + 4024*x^5 + 22912*x^6 + 134784*x^7 + 813476*x^8 + 5010904*x^9 + 31379808*x^10 +..+ A268302(n)*x^n +...

%e and J(x) = Product_{n>=1} (1-x^n) * (1 + x^n/A(x)) * (1 + x^(n-1)*A(x)).

%e Conjecture: Product_{n>=1} (1-x^n) * (1 + k*x^n/A(x)) * (1 + k*x^(n-1)*A(x)) yields an integer series for all integer k.

%t (* Calculation of constants {d,c}: *) {4/r, 1/(2*Sqrt[2*Pi]) * Sqrt[(r*s^4* Log[r]*(((-1 + s)*(-1 + r*s) * QPolyGamma[0, 1, r])/(r*s^2) - ((-1 + s)*(-1 + r*s)*Log[r] * Derivative[0, 1][QPochhammer][r, r])/(s^2 * QPochhammer[r, r]) + r*Log[r]*QPochhammer[r, r]*QPochhammer[s, r] * Derivative[0, 1][QPochhammer][1/(r*s), r] + ((-1 + s)*(QPochhammer[s, r]*(Log[r] + (1 - r*s)* QPolyGamma[0, -Log[r*s]/Log[r], r]) + r*(1 - r*s)*Log[r]* Derivative[0, 1][QPochhammer][s, r]))/(r*s^2 * QPochhammer[s, r]))) / (2*Log[r]^2 + (-3 + s*(1 + r + r*s)) * Log[r] * QPolyGamma[0, Log[s]/Log[r], r] + (-1 + s)*(-1 + r*s) * QPolyGamma[0, Log[s]/Log[r], r]^2 + ((3 - s*(1 + r + r*s))*Log[r] - 2*(-1 + s)*(-1 + r*s) * QPolyGamma[0, Log[s]/Log[r], r]) * QPolyGamma[0, -Log[r*s]/Log[r], r] + (-1 + s)*(-1 + r*s) * QPolyGamma[0, -Log[r*s]/Log[r], r]^2 + (-1 + s)*(-1 + r*s)* QPolyGamma[1, Log[s]/Log[r], r] + (-1 + s)*(-1 + r*s)* QPolyGamma[1, -Log[r*s]/Log[r], r])]} /. FindRoot[{(1 - 1/(r*s))*(1 - s)/(QPochhammer[r] * QPochhammer[1/(r*s), r] * QPochhammer[s, r]) == s, (-2 + s + r*s)*Log[r] + (-1 + s)*(-1 + r*s)*(QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s]/Log[r], r]) == 0}, {r, 1/7}, {s, 4}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 19 2024 *)

%o (PARI) {a(n) = my(A=2+x,t=floor(sqrt(2*n+1)+1/2)); for(i=0,n, A = (A + 1/sum(m=-t,t, x^(m*(m+1)/2) * (-A)^m +x*O(x^n)) )/2 ); 4^n/2 * polcoeff(A,n)}

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

%Y Cf. A268301, A268302, A268299, A190791.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 25 2016