

A126130


a(n) = (n+1)^n  n!.


4



1, 7, 58, 601, 7656, 116929, 2092112, 43006401, 999637120, 25933795801, 742968453888, 23297606120881, 793708546233344, 29192838847099425, 1152920196932478976, 48661170952876980481, 2185911204051268435968
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Fit a polynomial f of degree n1 to the first n nth powers of positive integers. Then a(n) = f(n+1). It is not necessary to actually determine the polynomial f; a(n) can be found by considering differences.
a(n1) is also the number of labeled rooted trees on n objects that are not increasing; i.e., at least one node has a label smaller than its parent's label. a(n) is the number of partial functions on n labeled objects that are not permutations.  Franklin T. AdamsWatters, Dec 25 2006
Equal to the number of partial functions [n]>[n] which are not permutations (equivalently, the number of nonsurjective partial functions [n]>[n]); i.e. equal to the cardinality of the complement PT_n\S_n where PT_n and S_n denote the partial transformation semigroup and symmetric group on [n].  James East, May 03 2007


LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..386


FORMULA

The polynomial f is equal to Sum_{k=1}^n s(n+1,k) x^{k1}, where the s(n,k) are the Stirling numbers of the first kind (A008275).  Franklin T. AdamsWatters, Dec 25 2006
E.g.f.: 1/(1  x)  LambertW(x)/(x*(1 + LambertW(x))), where LambertW() is the Lambert Wfunction.  Ilya Gutkovskiy, Aug 22 2018


EXAMPLE

The quadratic that fits (1,1), (2,8) and (3,27) is f(n) = 6n^211n+6. Then a(3) = f(4) = 58.


MATHEMATICA

Table[(n+1)^nn!, {n, 30 }] (* Harvey P. Dale, Jun 06 2015 *)


PROG

(PARI) vector(18, n, (n+1)^nn!)
(MAGMA) [(n+1)^n  n!: n in [1..20]]; // Altug Alkan, Mar 19 2018


CROSSREFS

Cf. A000169, A000142.
Sequence in context: A163048 A318233 A192940 * A323254 A123766 A005332
Adjacent sequences: A126127 A126128 A126129 * A126131 A126132 A126133


KEYWORD

easy,nonn


AUTHOR

Nick Hobson (nickh(AT)qbyte.org), Dec 18 2006


STATUS

approved



