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A166990 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.

%I A166990 #14 Oct 29 2024 12:57:12
%S A166990 1,2,7,30,147,786,4472,26644,164477,1044258,6782484,44887236,
%T A166990 301782361,2056250570,14172792355,98667874038,692948001906,
%U A166990 4904403499992,34951124337300,250617829087656,1807055528439771,13095146839953030
%N A166990 G.f.: A(x) = exp( Sum_{n>=1} A000172(n)*x^n/n ) where Franel number A000172(n) = Sum_{k=0..n} C(n,k)^3.
%C A166990 Analogous to the square of the g.f. of Catalan numbers (A000108):
%C A166990 C(x)^2 = exp( Sum_{n>=1} A000984(n)*x^n/n ) where central binomial coefficient A000984(n) = Sum_{k=0..n} C(n,k)^2.
%H A166990 G. C. Greubel, <a href="/A166990/b166990.txt">Table of n, a(n) for n = 0..500</a>
%F A166990 Self-convolution of A166991.
%F A166990 a(n) ~ c * 8^n / n^2, where c = 0.58462945... - _Vaclav Kotesovec_, Nov 27 2017, updated Oct 29 2024
%e A166990 G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 147*x^4 + 786*x^5 + 4472*x^6 +...
%e A166990 log(A(x)) = 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 A166990 a[n_] := Sum[(Binomial[n, k])^3, {k, 0, n}]; f[x_] := Sum[a[n]*x^n/n, {n, 1, 75}]; CoefficientList[Series[Exp[f[x]], {x, 0, 50}], x] (* _G. C. Greubel_, May 30 2016 *)
%t A166990 nmax = 30; Clear[a]; franel = RecurrenceTable[{n^2*a[n] == (7*n^2 - 7*n + 2)*a[n-1] + 8*(n-1)^2*a[n-2], a[1] == 2, a[2] == 10}, a, {n, 1, nmax}]; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[franel[[k]]*a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* _Vaclav Kotesovec_, Oct 27 2024 *)
%o A166990 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,sum(k=0,m,binomial(m,k)^3)*x^m/m)+x*O(x^n)),n)}
%Y A166990 Cf. A000172 (Franel numbers), A166991, A166992, A218117, A218119.
%K A166990 nonn
%O A166990 0,2
%A A166990 _Paul D. Hanna_, Nov 17 2009