A032037 Doubles (index 2+) under "AIJ" (ordered, indistinct, labeled) transform.

1, 2, 18, 264, 5400, 141840, 4551120, 172529280, 7545363840, 373944211200, 20711190931200, 1267784551756800, 84991791159475200, 6193091146059417600, 487361761916020992000, 41192820513212239872000, 3721763273059549605888000, 357950394802026289815552000

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 92.
• Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Programs

Mathematica

a[1]=1; a[2]=2; a[n_] := a[n]=3(2n-3)a[n-1]-(n-1)(n-3)a[n-2]

PARI

Vec(serlaplace(serreverse(x*(1-2*x)/(1-x) + O(x^20)))) \\ Andrew Howroyd, Sep 19 2018

Formula

a(n) = n!*A001003(n-1). - Vladeta Jovovic, Dec 06 2002

E.g.f.: series reversion of x*(1-2*x)/(1-x). - Andrew Howroyd, Sep 19 2018

Assuming offset = 0:

a(n) = Sum_{k=0..n} Sum{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!. - Peter Luschny, Sep 26 2022

Extensions

Terms a(17) and beyond from Andrew Howroyd, Sep 19 2018