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A206591 E.g.f.: Sum_{n>=0} x^(n^2)*exp(n^2*x). 2
1, 1, 2, 3, 28, 485, 5766, 53767, 430088, 3459465, 53303050, 1746391691, 58977262092, 1706810202253, 42923448632078, 965348202349455, 19877420584519696, 385436337079476497, 7654870637722391058, 199927590326456092435, 8556099311090244142100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare to the partial theta series identity:

Sum_{n>=0} x^(n^2) = Sum_{n>=0} x^n * Product_{k=1..n} (1-x^(4*k-3))/(1-x^(4*k-1)).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

FORMULA

E.g.f.: Sum_{n>=0} x^n*exp(n*x) * Product_{k=1..n} (1 - x^(4*k-3)*exp((4*k-3)*x))/(1 - x^(4*k-1)*exp((4*k-1)*x)).

Let q = x*exp(x), then the e.g.f. equals the continued fraction:

A(x) = 1/(1- q/(1- q*(q^2-1)/(1- q^5/(1- q^3*(q^4-1)/(1- q^9/(1- q^5*(q^6-1)/(1- q^13/(1- q^7*(q^8-1)/(1- ...))))))))), due to a partial elliptic theta function identity.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! + 28*x^4/4! + 485*x^5/5! +...

where the e.g.f. is defined by:

A(x) = 1 + x*exp(x) + x^4*exp(4*x) + x^9*exp(9*x) + x^16*exp(16*x) +...

Let q = x*exp(x), then the e.g.f. also equals the q-series:

A(x) = 1 + q*(1-q)/(1-q^3) + q^2*(1-q)*(1-q^5)/((1-q^3)*(1-q^7)) + q^3*(1-q)*(1-q^5)*(1-q^9)/((1-q^3)*(1-q^7)*(1-q^11)) +...

PROG

(PARI) {a(n)=n!*polcoeff(sum(m=0, sqrtint(n+1), x^(m^2)*exp(m^2*x+x*O(x^n))), n)}

(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(1+sum(m=1, n+1, x^m*exp(m*X)*prod(k=1, m, (1 - x^(4*k-3)*exp((4*k-3)*X))/(1 - x^(4*k-1)*exp((4*k-1)*X))) ), n)}

for(n=0, 35, print1(a(n), ", "))

CROSSREFS

Cf. A193421, A206592.

Sequence in context: A126266 A219975 A319146 * A003017 A096580 A324941

Adjacent sequences:  A206588 A206589 A206590 * A206592 A206593 A206594

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 09 2012

STATUS

approved

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Last modified October 27 00:02 EDT 2021. Contains 348270 sequences. (Running on oeis4.)