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%I #15 Jul 02 2019 18:40:20
%S 1,1,4,27,280,4025,75876,1800253,52193408,1807302897,73406128420,
%T 3446236588421,184750419871920,11194423784630281,759960096829452260,
%U 57367378069894391325,4783586470578255085696,438054092182322814028001,43827052650093379145736900
%N E.g.f.: Sum_{n>=0} x^n * exp(n*(n+1)/2*x).
%H Vaclav Kotesovec, <a href="/A193467/b193467.txt">Table of n, a(n) for n = 0..300</a>
%F 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.
%F Let q = exp(x), then the e.g.f. equals the continued fraction:
%F 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.
%F O.g.f.: Sum_{k>=0} k! * x^k / (1 - binomial(k+1,2)*x)^(k+1). - _Ilya Gutkovskiy_, Jul 02 2019
%e 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! +...
%e where
%e A(x) = 1 + x*exp(x) + x^2*exp(3*x) + x^3*exp(6*x) + x^4*exp(10*x) +...
%e By a q-series identity:
%e 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))) +...
%o (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)}
%o (PARI) /* q-series identity: */
%o {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)}
%Y Cf. A193421, A193469, A177385, A193466.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 27 2011
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