A093964 a(n) = Sum_{k=1..n} k*k!*C(n,k).

0, 1, 6, 33, 196, 1305, 9786, 82201, 767208, 7891281, 88776910, 1085051121, 14322674796, 203121569833, 3080677142466, 49764784609065, 853110593298256, 15469738758475041, 295858753755835158, 5951981987323272001, 125652953065713520020, 2777591594084193600441

Offset

0,3

Comments

Limit to which the columns of array A093966 converge.

Number of objects in all permutations of n objects taken 1,2,...,n at a time. Example: a(2)=6 because the permutations of {a,b} taken 1 and 2 at a time are: a,b,ab and ba, containing altogether 1+1+2+2=6 objects. a(n)=Sum(k*A008279(n,k),k=1..n). - Emeric Deutsch, Aug 16 2006

The number of sequences -where each member is an element in a set consisting of n elements- such that the last member is a repetition of a former member. Example: Set of possible members: {l,r}. Sequences such that the last member is a repetition of a former member: l,l; r,r; l,r,l; l,r,r; r,l,l; r,l,r. a(n)=Sum(k*A008279(n,k),k=1..n). [From Franz Fritsche (ff(AT)simple-line.de), Feb 22 2009]

The total number of elements in all ascending runs (including runs of length 1) over all permutations of {1,2,...,n}. a(2) = 6 because in the permutations [1,2] and [2,1] there are 4 runs of length 1 and 1 run of length 2. a(n) = Sum_{k>=1} A132159(n,k)*k. - Geoffrey Critzer, Feb 24 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

E.g.f.: x*exp(x)/(1-x)^2. - Vladeta Jovovic, Apr 24 2004

a(n) = 1 + (n-1)*floor(e*n!) = 1 + (n-1)*A000522(n) = A000522(n+1) - 2*A000522(n) = A001339(n) - A000522(n). - Henry Bottomley, Dec 22 2008

a(n) = n if n < 2, a(n) = n*((n+1)/(n-1)*a(n-1) - a(n-2)) for n >= 2. - Alois P. Heinz, Jan 21 2013

E.g.f.: x*(1- 12*x/(Q(0)+6*x-3*x^2))/(1-x)^2, where Q(k) = 2*(4*k+1)*(32*k^2+16*k+x^2-6) - x^4*(4*k-1)*(4*k+7)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013

G.f.: conjecture: T(0)/x - 1/x, where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013

a(n) = n*a(n-1) + A007526(n), a(0) = 0. - David M. Cerna, May 12 2014

Example

G.f. = x + 6*x^2 + 33*x^3 + 196*x^4 + 1305*x^5 + 9786*x^6 + 82201*x^7 + ...

Maple

seq(add(k*n!/(n-k)!, k=1..n), n=0..20); # Emeric Deutsch, Aug 16 2006
# second Maple program:
a:= proc(n) a(n):=`if`(n<2, n, n*((n+1)/(n-1)*a(n-1)-a(n-2))) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 21 2013

Mathematica

nn=21; Range[0, nn]!CoefficientList[Series[D[Exp[y x]/(1-x)^2, y]/.y->1, {x, 0, nn}], x] (* Geoffrey Critzer, Feb 24 2014 *)

Prog

(PARI) a(n)=sum(k=1, n, k*k!*binomial(n, k))
(Magma) [0] cat [n le 2 select 6^(n-1) else n*((n+1)*Self(n-1) - (n-1)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Dec 29 2021
(Sage) [factorial(n)*( x*exp(x)/(1-x)^2 ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Dec 29 2021

Crossrefs

Cf. A000522, A001339, A007526, A008279.

Row n=2 of A210472. - Alois P. Heinz, Jan 23 2013

Sequence in context: A286187 A260774 A218182 * A361776 A193665 A097662

Adjacent sequences: A093961 A093962 A093963 * A093965 A093966 A093967

Keyword

nonn

Author

Ralf Stephan, Apr 20 2004

Extensions

a(0) inserted by Alois P. Heinz, Jan 21 2013

Status

approved