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A268300 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. 5
1, 7, 119, 2118, 42523, 914922, 20745494, 487390092, 11764545555, 289962708802, 7267069560834, 184626340341588, 4744080078088734, 123075608359376932, 3219261610951795084, 84806249132678044440, 2248017950109054256899, 59917503707743905031346, 1604813748929693765997450, 43170742498490205711682564, 1165893490887496323343495146, 31598783791475055433157814444, 859179326846115018832395000820 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

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

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

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

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 29.10159109069361717048796233905065832... and c = 0.57417747020768285925989822148605305... . - Vaclav Kotesovec, Mar 02 2016

EXAMPLE

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 +...

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

-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)) *...

also

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)) *...).

RELATED SERIES.

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 +...

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.

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

then J(x) is an integer series:

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 +...

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

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.

PROG

(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)}

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

CROSSREFS

Cf. A268301, A268302, A268299, A190791.

Sequence in context: A057769 A221031 A221323 * A304917 A113667 A192565

Adjacent sequences:  A268297 A268298 A268299 * A268301 A268302 A268303

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 25 2016

STATUS

approved

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Last modified May 17 03:25 EDT 2022. Contains 353727 sequences. (Running on oeis4.)