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.

1, 2, 21, 504, 21380, 1405800, 132139140, 16801276800, 2775758497344, 577868994460800, 147973478687496000, 45703277816543424000, 16753246307626306832640, 7190163806348621417679360, 3571395525388698501285792000

Offset: 1

Author

Paul D. Hanna, May 24 2013

Keywords

• nonn

Comments

Name is directly from a formula for A129505 given by Vladimir Kruchinin.

Examples

			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 +...
where
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! +...
and
-log(1-A(x)) = 2*x/2! + 6*x^2/3! + 34*x^3/4! + 280*x^4/5! + 3013*x^5/6! +...
		

Links

• D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.

Crossrefs

Cf. A204248, A129505.

Programs

PARI

{a(n)=polcoeff(prod(k=0, 2*n-2, 1+k*x), n-1)*n!^2*(n-1)!/(2*n-1)!}

Python

from math import factorial, comb
from sympy.functions.combinatorial.numbers import stirling
def A226057(n): return factorial(n)*stirling(m:=(n<<1)-1,n,kind=1)//comb(m,n-1) # Chai Wah Wu, Jun 08 2025

Formula

a(n) = n!^2*(n-1)!/(2*n-1)! * {[x^(n-1)] Product_{k=0..2*n-2} (1+k*x)}.

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.

a(n) = n*A204248(n-1), where A204248(n) = permanent of the n-th principal submatrix of A002024.