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A166991 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/(2*n) ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

%I A166991 #10 Nov 27 2017 09:44:18
%S A166991 1,1,3,12,57,300,1693,10045,61890,392688,2550843,16891566,113660475,
%T A166991 775223595,5349057132,37280705406,262119009927,1857241951359,
%U A166991 13250054817027,95110710932424,686490953423700,4979704242810870
%N A166991 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/(2*n) ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
%H A166991 G. C. Greubel, <a href="/A166991/b166991.txt">Table of n, a(n) for n = 0..500</a>
%F A166991 Self-convolution yields A166990.
%F A166991 a(n) ~ c * 8^n / n^2, where c = 0.231776... - _Vaclav Kotesovec_, Nov 27 2017
%e A166991 G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1693*x^6 +...
%e A166991 log(A(x)^2) = 2*x + 10*x^2/2 + 56*x^3/3 + 346*x^4/4 + 2252*x^5/5 + 15184*x^6/6 + 104960*x^7/7 +...+ A000172(n)*x^n/n +...
%t A166991 a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/(2*n), {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* _G. C. Greubel_, May 30 2016 *)
%o A166991 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)/2*x^m/m)+x*O(x^n)),n)}
%Y A166991 Cf. A000172 (Franel numbers), A166990, A166993, A218118, A218120.
%K A166991 nonn
%O A166991 0,3
%A A166991 _Paul D. Hanna_, Nov 17 2009