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

Author

Len Smiley

Keywords

• nonn
• easy

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

Crossrefs

Cf. A003149, A064570, A229464.

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

Programs

Mathematica

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

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

a(n) = 3F0(1,1,-n;;-1). - R. J. Mathar, Jun 26 2024

Extensions

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