cp's OEIS Frontend

A129505 Number of permutations of 2n-1 objects with exactly n cycles.

%I A129505 #66 Jun 09 2025 00:52:12
%S A129505 1,3,35,735,22449,902055,44990231,2681453775,185953177553,
%T A129505 14710753408923,1307535010540395,129006659818331295,
%U A129505 13990945200239106865,1654339178844590073615,211821088794711294496815,29197210605623737977801375,4310704065427058593776844065
%N A129505 Number of permutations of 2n-1 objects with exactly n cycles.
%H A129505 Vincenzo Librandi, <a href="/A129505/b129505.txt">Table of n, a(n) for n = 1..200</a>
%H A129505 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 A129505 Unsigned central Stirling numbers of the first kind:
%F A129505 G.f.: A(x) = Sum_{n>=0} a(n)*(2*n-1)!/n!*x^n = B'(x), where B(x) satisfies B(x)^2 = x*log(1/(1-B(x))). - _Vladimir Kruchinin_, Jun 10 2012
%F A129505 a(n) = ((2*n+1)*(-1)^n*((Sum_{i=1..n-1} (Stirling1(2*i-1,i)*C(2*n,2*i-1)*Stirling1(2*(n-i)+1,n-i))/((n-i)*C(n,i)))-n*Stirling1(2*n-1,n) + Stirling1(2*n,n)))/(n+1). - _Vladimir Kruchinin_, Feb 28 2013
%F A129505 a(n) ~ (1+2*c)/(8*c*sqrt(Pi*(-1-c))) * (-8*c^2/(exp(1)*(1+2*c)))^n * n^(n-3/2), where c = LambertW(-1,-1/(2*exp(1/2))). - _Vaclav Kotesovec_, Dec 28 2013
%F A129505 a(n) = abs(C(2*n-1,n-1)*Sum_{i=1..n-1} (Stirling1(n-1,n-i-1)*Stirling1(n,i+1)/C(n-1,i))). - _Chai Wah Wu_, Jun 08 2025
%t A129505 t[n_] := SymmetricPolynomial[n, Range[1, 2 n]]
%t A129505 Table[t[n], {n, 1, 6}]  (* A129505 *)
%t A129505 (* _Clark Kimberling_, Dec 30 2011 *)
%t A129505 Table[Abs[StirlingS1[2*n-1, n]], {n, 1, 20}] (* _Vaclav Kotesovec_, Dec 28 2013 *)
%o A129505 (PARI) a(n)=polcoeff(prod(k=0,2*n-2,1+k*x),n-1)
%o A129505 (PARI) vector(66, n, abs( stirling(2*n-1, n, 1) ) ) /* _Joerg Arndt_, Jun 09 2012 */
%o A129505 (Maxima)
%o A129505 a(n):=((2*n+1)*(-1)^n*((sum((stirling1(2*i-1,i)*binomial(2*n,2*i-1)* stirling1(2*(n-i)+1,n-i))/((n-i)*binomial(n,i)),i,1,n-1)) -n*stirling1(2*n-1,n) +stirling1(2*n,n)))/(n+1); /* _Vladimir Kruchinin_, Feb 28 2013 */
%o A129505 (Maxima) a(n):=coeff(expand(product(x+i,i,1,2*(n-1))),x,(n-1)); /* _Lorraine Lee_, Oct 12 2019 */
%o A129505 (Haskell)
%o A129505 a129505 n = abs $ a008275 (2 * n - 1) n -- _Reinhard Zumkeller_, Mar 02 2014
%o A129505 (Python)
%o A129505 from sympy.functions.combinatorial.numbers import stirling
%o A129505 def A129505(n): return stirling((n<<1)-1,n,kind=1) # _Chai Wah Wu_, Jun 08 2025
%Y A129505 Cf. A008275, A129506.
%Y A129505 Cf. A238685.
%K A129505 nonn
%O A129505 1,2
%A A129505 _Paul D. Hanna_, Apr 18 2007
%E A129505 Minor edits by _Vaclav Kotesovec_, Mar 31 2014