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

0,3

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

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

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 *)

Crossrefs

Cf. A000262, A000290, A001477, A206703.

Sequence in context: A231482 A367233 A122827 * A009018 A289996 A335344

Adjacent sequences: A103191 A103192 A103193 * A103195 A103196 A103197

Keyword

easy,nonn

Author

Vladeta Jovovic, Mar 18 2005

Status

approved