A115327 E.g.f.: exp(x + 3/2*x^2).

1, 1, 4, 10, 46, 166, 856, 3844, 21820, 114076, 703216, 4125496, 27331624, 175849480, 1241782816, 8627460976, 64507687696, 478625814544, 3768517887040, 29614311872416, 244419831433696, 2021278543778656, 17419727924101504

Offset

0,3

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_{3n} whose disjoint cycle decomposition consists of n cycles of length 3. - Luis Manuel Rivera Martínez, Feb 26 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

John Campbell, A class of symmetric difference-closed sets related to commuting involutions, Discrete Mathematics & Theoretical Computer Science, Vol 19 no. 1, 2017.

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).

Formula

Term-by-term square equals A115328 which has e.g.f.: exp(x/(1-3*x))/sqrt(1-9*x^2).

From Paul Barry, Apr 10 2009: (Start)

G.f.: 1/(1-x-3*x^2/(1-x-6*x^2/(1-x-9*x^2/(1-x-12*x^2/(1-... (continued fraction);

a(n) = a(n-1)+3*(n-1)*a(n-2). (End)

a(n) ~ exp(sqrt(n/3)-n/2-1/12)*3^(n/2)*n^(n/2)/sqrt(2). - Vaclav Kotesovec, Oct 19 2012

G.f.: 1/Q(0), where Q(k)= 1 + 3*x*k - x/(1 - 3*x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 17 2013

a(n) = n!*Sum_{k=0..floor(n/2)}3^k/(2^k*k!*(n-2*k)!). - Luis Manuel Rivera Martínez, Feb 26 2015

Mathematica

Range[0, 20]! CoefficientList[Series[Exp[(x + 3 / 2 x^2)], {x, 0, 20}], x] (* Vincenzo Librandi, May 22 2013 *)

Prog

(PARI) a(n)=local(m=3); n!*polcoeff(exp(x+m/2*x^2+x*O(x^n)), n)

Crossrefs

Column k=3 of A359762.

Cf. A115328.

Sequence in context: A149228 A149229 A149230 * A233395 A197664 A099606

Adjacent sequences: A115324 A115325 A115326 * A115328 A115329 A115330

Keyword

nonn

Author

Paul D. Hanna, Jan 20 2006

Status

approved