A103194 LAH transform of squares.

0, 1, 6, 39, 292, 2505, 24306, 263431, 3154824, 41368977, 589410910, 9064804551, 149641946796, 2638693215769, 49490245341642, 983607047803815, 20646947498718736, 456392479671188001, 10595402429677269174, 257723100178182605287, 6553958557721713088820

Offset: 0

Author

Vladeta Jovovic, Mar 18 2005

Keywords

• easy
• nonn

Comments

If the e.g.f. of b(n) is E(x) and a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*b(k), then the e.g.f. of a(n) is E(x/(1-x))/(1-x). - Vladeta Jovovic, Apr 16 2005

a(n) is the total number of elements in all partial permutations (injective partial functions) of {1,2,...,n} that are in a cycle. A fixed point is considered to be in a cycle. a(n) = Sum_{k=0..n} A206703(n,k)*k. - Geoffrey Critzer, Feb 11 2012

a(n) is the total number of elements in all partial permutations (injective partial functions) of {1,2,...,n} that are undefined, i.e., they do not have an image.- Geoffrey Critzer, Feb 09 2022

a(n) is the total length of all increasing subsequences over all n-permutations. Cf. A002720. - Geoffrey Critzer, Feb 09 2022

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000262, A000290, A001477, A206703.

Programs

Maple

with(combstruct): SetSeqSetL := [T, {T=Set(S), S=Sequence(U, card >= 1), U=Set(Z, card=1)}, labeled]: seq(k*count(SetSeqSetL, size=k), k=0..18); # Zerinvary Lajos, Jun 06 2007
a := n -> n!*hypergeom([2, 1-n], [1, 1], -1):
seq(simplify(a(n)),n=0..20); # Peter Luschny, Mar 30 2015

Mathematica

nn = 20; a = 1/(1 - x); ay = 1/(1 - y x); D[Range[0, nn]! CoefficientList[ Series[Exp[a x] ay, {x, 0, nn}], x], y] /. y -> 1  (* Geoffrey Critzer, Feb 11 2012 *)

Formula

a(n) = Sum_{k=0..n} (n!/k!)*binomial(n-1, k-1)*k^2.

E.g.f.: x/(1-x)^2*exp(x/(1-x)).

Recurrence: (n-1)*a(n) - n*(2*n-1)*a(n-1) + n*(n-1)^2*a(n-2) = 0.

a(n) = n*A000262(n). - Vladeta Jovovic, Mar 20 2005

a(n) ~ n! * exp(-1/2 + 2*sqrt(n))*n^(1/4)/(2*sqrt(Pi)). - Vaclav Kotesovec, Aug 13 2013

a(n) = n!*hypergeom([2, 1-n], [1, 1], -1). - Peter Luschny, Mar 30 2015

a(n) = Sum_{k=1..n} k*(n-k)!*binomial(n,k)^2. - Ridouane Oudra, Jun 17 2025