A115331 E.g.f.: exp(x+5/2*x^2).

1, 1, 6, 16, 106, 426, 3076, 15856, 123516, 757756, 6315976, 44203776, 391582456, 3043809016, 28496668656, 241563299776, 2378813448976, 21703877431056, 223903020594016, 2177251989389056, 23448038945820576, 241173237884726176

Offset: 0

Author

Paul D. Hanna, Jan 20 2006

Keywords

• nonn

Comments

Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.

a(n) is also the number of square roots of any permutation in S_{5n} whose disjoint cycle decomposition consists of n cycles of length 5. - Luis Manuel Rivera Martínez, Feb 26 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
• Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations , Australas. J. Combin. 52 (2012), 41-54 (Theorems 1 and 2).

Crossrefs

Column k=5 of A359762.

Cf. A115332.

Programs

Mathematica

Range[0, 20]! CoefficientList[Series[Exp[(x + 5 / 2 x^2)], {x, 0, 20}], x] (* Vincenzo Librandi, May 22 2013 *)

PARI

a(n)=local(m=5);n!*polcoeff(exp(x+m/2*x^2+x*O(x^n)),n)

Formula

Term-by-term square equals A115332 which has e.g.f.: exp(x/(1-5*x))/sqrt(1-25*x^2).

a(n) ~ exp(sqrt(n/5)-n/2-1/20)*5^(n/2)*n^(n/2)/sqrt(2). - Vaclav Kotesovec, Oct 19 2012

a(n) = n!*Sum_{k=0..floor(n/2)}5^k/(2^k*k!*(n-2*k)!). - Luis Manuel Rivera Martínez, Feb 26 2015

O.g.f.: 1/(1-x - 5*x^2/(1-x - 10*x^2/(1-x - 15*x^2/(1-x - 20*x^2/(1-x - 25*x^2/(1-x -...)))))), a continued fraction (after Paul Barry in A115327). - Paul D. Hanna, Mar 08 2015