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A226057 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.

%I A226057 #18 Jun 09 2025 00:59:46
%S A226057 1,2,21,504,21380,1405800,132139140,16801276800,2775758497344,
%T A226057 577868994460800,147973478687496000,45703277816543424000,
%U A226057 16753246307626306832640,7190163806348621417679360,3571395525388698501285792000
%N A226057 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 A226057 Name is directly from a formula for A129505 given by _Vladimir Kruchinin_.
%H A226057 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 A226057 a(n) = n!^2*(n-1)!/(2*n-1)! * {[x^(n-1)] Product_{k=0..2*n-2} (1+k*x)}.
%F A226057 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 A226057 a(n) = n*A204248(n-1), where A204248(n) = permanent of the n-th principal submatrix of A002024.
%e A226057 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 A226057 where
%e A226057 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 A226057 and
%e A226057 -log(1-A(x)) = 2*x/2! + 6*x^2/3! + 34*x^3/4! + 280*x^4/5! + 3013*x^5/6! +...
%o A226057 (PARI) {a(n)=polcoeff(prod(k=0, 2*n-2, 1+k*x), n-1)*n!^2*(n-1)!/(2*n-1)!}
%o A226057 (Python)
%o A226057 from math import factorial, comb
%o A226057 from sympy.functions.combinatorial.numbers import stirling
%o A226057 def A226057(n): return factorial(n)*stirling(m:=(n<<1)-1,n,kind=1)//comb(m,n-1) # _Chai Wah Wu_, Jun 08 2025
%Y A226057 Cf. A204248, A129505.
%K A226057 nonn
%O A226057 1,2
%A A226057 _Paul D. Hanna_, May 24 2013