|
%I #12 Jan 23 2019 10:09:55
%S 1,2,21,504,21380,1405800,132139140,16801276800,2775758497344,
%T 577868994460800,147973478687496000,45703277816543424000,
%U 16753246307626306832640,7190163806348621417679360,3571395525388698501285792000
%N E.g.f. A(x) satisfies: A(x)^2 = -x*log(1-A(x)) where A(x) = Sum_{n>=1} a(n)*x^n/n!^2.
%C Name is directly from a formula for A129505 given by _Vladimir Kruchinin_.
%H D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
%F a(n) = n!^2*(n-1)!/(2*n-1)! * {[x^(n-1)] Product_{k=0..2*n-2} (1+k*x)}.
%F a(n) = n!^2*(n-1)!/(2*n-1)! * A129505(n), where A129505(n) = number of permutations of 2n-1 objects with exactly n cycles.
%F a(n) = n*A204248(n-1), where A204248(n) = permanent of the n-th principal submatrix of A002024.
%e E.g.f.: A(x) = x + 2*x^2/2!^2 + 21*x^3/3!^2 + 504*x^4/4!^2 + 21380*x^5/5!^2 +...
%e where
%e A(x)^2 = 2*x^2/2! + 6*x^3/3! + 34*x^4/4! + 280*x^5/5! + 3013*x^6/6! + 39963*x^7/7! + 629541*x^8/8! +...
%e and
%e -log(1-A(x)) = 2*x/2! + 6*x^2/3! + 34*x^3/4! + 280*x^4/5! + 3013*x^5/6! +...
%o (PARI) {a(n)=polcoeff(prod(k=0, 2*n-2, 1+k*x), n-1)*n!^2*(n-1)!/(2*n-1)!}
%Y Cf. A204248, A129505.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 24 2013
| 