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

1,2

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.

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)

Example

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

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 *)

Prog

(Sage) [binomial(n, 4)*factorial (n-3) for n in range(4, 21)] # Zerinvary Lajos, Jul 07 2009
(Magma) [n*Factorial(n+3)/24: n in [1..20]]; // Vincenzo Librandi, Oct 11 2011
(PARI) a(n) = n*(n+3)!/24; \\ Altug Alkan, Oct 08 2017

Crossrefs

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

Cf. A001620, A091725, A099285.

Sequence in context: A003952 A252703 A033136 * A199527 A137684 A097394

Adjacent sequences: A061203 A061204 A061205 * A061207 A061208 A061209

Keyword

nonn

Author

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

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

Status

approved