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.

1, 1, 3, 12, 57, 300, 1693, 10045, 61890, 392688, 2550843, 16891566, 113660475, 775223595, 5349057132, 37280705406, 262119009927, 1857241951359, 13250054817027, 95110710932424, 686490953423700, 4979704242810870

Offset: 0

Author

Paul D. Hanna, Nov 17 2009

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1693*x^6 +...
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 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000172 (Franel numbers), A166990, A166993, A218118, A218120.

Programs

Mathematica

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 *)

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)}

Formula

Self-convolution yields A166990.

a(n) ~ c * 8^n / n^2, where c = 0.231776... - Vaclav Kotesovec, Nov 27 2017