A061206 a(n) = total number of occurrences of the consecutive pattern 1324 in all permutations of [n+3].

1, 10, 90, 840, 8400, 90720, 1058400, 13305600, 179625600, 2594592000, 39956716800, 653837184000, 11333177856000, 207484333056000, 4001483566080000, 81096733605888000, 1723305589125120000, 38318206628782080000, 889833909490606080000, 21543347282404147200000

Offset: 1

Author

Melvin J. Knight (knightmj(AT)juno.com), May 30 2001

Keywords

• nonn

Comments

a(n) is the number of sequences of n+3 balls colored with at most n colors such that exactly four balls are the same color as some other ball in the sequence. - Jeremy Dover, Sep 27 2017

Examples

			a(4)=840 because 4*(7!)/24 = 4*7*6*5 = 840.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..300
• Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.

Crossrefs

Cf. A000142, A001286, A001339, A001563, A005990, A264173.

Cf. A001620, A091725, A099285.

Programs

Magma

[n*Factorial(n+3)/24: n in [1..20]]; // Vincenzo Librandi, Oct 11 2011

Maple

a := n -> n!*binomial(-n,4): seq(a(n),n=1..20); # Peter Luschny, Apr 29 2016

Mathematica

Array[# (# + 3)!/24 &, 20] (* or *) Array[#!*Binomial[-#, 4] &, 20] (* Michael De Vlieger, Sep 30 2017 *)

PARI

a(n) = n*(n+3)!/24; \\ Altug Alkan, Oct 08 2017

Sage

[binomial(n,4)*factorial (n-3) for n in range(4, 21)] # Zerinvary Lajos, Jul 07 2009

Formula

a(n) = n*(n+3)!/24.

If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i) * x^(k-j), then a(n-3) = (-1)^n*f(n,4,-2), (n >= 4). - Milan Janjic, Mar 01 2009

E.g.f.: x/(1-x)^5. (This was initiated by e-mail exchange with Gary Detlefs.) - Wolfdieter Lang, May 28 2010

a(n) = ((n+4)!/6) * Sum_{k=1..n} (k+2)!/(k+4)!. - Gary Detlefs, Aug 05 2010

a(n) = Sum_{k>0} k * A264173(n+3,k). - Alois P. Heinz, Nov 06 2015

a(n) = n!*binomial(-n,4). - Peter Luschny, Apr 29 2016

From Amiram Eldar, Sep 24 2022: (Start)

Sum_{n>=1} 1/a(n) = 118/3 - 16*e - 4*gamma + 4*Ei(1), where gamma is Euler's constant (A001620) and Ei(1) is the exponential integral at 1 (A091725).

Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 - 8/e + 4*gamma - 4*Ei(-1), where -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

Extensions

More terms from Jason Earls, Jun 12 2001

Corrected by Zerinvary Lajos, Jul 07 2009

More precise definition from Alois P. Heinz, Nov 06 2015