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A193467 E.g.f.: Sum_{n>=0} x^n * exp(n*(n+1)/2*x). 4
1, 1, 4, 27, 280, 4025, 75876, 1800253, 52193408, 1807302897, 73406128420, 3446236588421, 184750419871920, 11194423784630281, 759960096829452260, 57367378069894391325, 4783586470578255085696, 438054092182322814028001, 43827052650093379145736900 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..18.

FORMULA

E.g.f.: A(x) = Sum_{n>=0} x^n*exp(n*x)*Product_{k=1..n} (1 - x*exp((2*k-1)*x)) / (1 - x*exp(2*k*x)), due to a q-series identity.

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

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

EXAMPLE

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 280*x^4/4! + 4025*x^5/5! + 75876*x^6/6! + 1800253*x^7/7! +...

where

A(x) = 1 + x*exp(x) + x^2*exp(3*x) + x^3*exp(6*x) + x^4*exp(10*x) +...

By a q-series identity:

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

PROG

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

(PARI) /* q-series identity: */

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*exp(m*x+x*O(x^n))*prod(k=1, m, (1-x*exp((2*k-1)*x+x*O(x^n)))/(1-x*exp((2*k)*x+x*O(x^n)))))); n!*polcoeff(A, n)}

CROSSREFS

Cf. A193421, A193469, A177385, A193466.

Sequence in context: A104653 A194787 A020558 * A179494 A203157 A119820

Adjacent sequences:  A193464 A193465 A193466 * A193468 A193469 A193470

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 27 2011

STATUS

approved

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Last modified April 16 18:13 EDT 2014. Contains 240625 sequences.