A046662 Sum of mistyped version of binomial coefficients.

1, 2, 7, 52, 749, 17686, 614227, 29354312, 1844279257, 147273109354, 14561325802271, 1745720380045852, 249461639720702917, 41886684733511640062, 8164388189339113521259, 1828191138807263097870256, 466057478369217965809683377, 134193343258948416556377786322

Offset

0,2

Comments

Origin of the name of this sequence: Binomial coefficients are n!/((n-k)!*k!) but if parentheses are omitted in the denominator, the formula might result in n!/(n-k)!*k! = n!*k!/(n-k)! and the sum giving a(n) instead of 2^n. If k! is forgotten altogether, one gets A000522. - Olivier Gérard, Mar 05 2024

Binomial transform of (n!)^2. - Peter Luschny, May 31 2014

Links

Seiichi Manyama, Table of n, a(n) for n = 0..253

Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From N. J. A. Sloane, Feb 06 2013

Formula

a(n) = Sum_{k=0..n} n!*k!/(n-k)!.

E.g.f.: exp(x)*F(x), with F(x) = Sum_{k>=0} k!*x^k. - Ralf Stephan, Apr 02 2004

a(n) = n^2*a(n - 1) - n*(n - 1)*a(n - 2) + 1. - Vladeta Jovovic, Jul 15 2004

From Peter Bala, Nov 26 2017: (Start)

a(k) == a(0) (mod k) for all k (by the inhomogeneous recurrence equation).

More generally, a(n+k) = a(n) (mod k) for all n and k (by an induction argument on n). It follows that for each positive integer k, the sequence a(n) (mod k) is periodic, with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, ... with exact period 10. (End)

G.f.: Sum_{k>=0} (k!)^2*x^k/(1 - x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019

a(n) ~ (n!)^2. - Vaclav Kotesovec, May 03 2021

Mathematica

Table[Sum[(n!k!)/(n-k)!, {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Sep 29 2012 *)

Crossrefs

Cf. A003149, A064570, A229464.

Cf. A000522 (Total number of ordered k-subsets of [1,n], k=0..n.)

Sequence in context: A138737 A216086 A210856 * A237195 A275597 A118191

Adjacent sequences: A046659 A046660 A046661 * A046663 A046664 A046665

Keyword

nonn,easy,changed

Author

Len Smiley

Extensions

Corrected and extended by Harvey P. Dale, Sep 29 2012

Status

approved