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A051295 a(0)=1; thereafter, a(m+1) = Sum_{k=0 to m} k!*a(m-k). 15
1, 1, 2, 5, 15, 54, 235, 1237, 7790, 57581, 489231, 4690254, 49986715, 585372877, 7463687750, 102854072045, 1522671988215, 24093282856182, 405692082526075, 7242076686885157, 136599856992122366, 2714409550073698925 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = number of permutations on [n] that contain a 132 pattern only as part of a 4132 pattern. For example, a(4) = 15 counts the 14 132-avoiding permutations on [4] (Catalan numbers A000108) and 4132.

a(n) is the number of permutations on [n] that contain a (scattered) 342 pattern only as part of a 1342 pattern. For example, 412635 fails because 463 is an offending 342 pattern (= 231 pattern).

This sequence gives the number of permutations of {1,2,...,n} such that the elements of each cycle of the permutation form an interval. - Michael Albert, Dec 14 2004

Starting (1, 2, 5, 15,...) = row sums of triangle A143965. - Gary W. Adamson, Apr 10 2009

Number of compositions of n where there are (k-1)! sorts of part k. - Joerg Arndt, Aug 04 2014

LINKS

Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 0..448 (first 100 terms from Vincenzo Librandi)

David Callan, A combinatorial interpretation of the eigensequence for composition, arXiv:math/0507169 [math.CO]

David Callan, A Combinatorial Interpretation of the Eigensequence for Composition, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.4.

Stefan Forcey, Aaron Lauve and Frank Sottile, Cofree compositions of coalgebras, Annals of Combinatorics 17 (1) pp. 105-130 March, 2013.

Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

L. Pudwell, Enumeratino schemes for permutations avoid abarred patterns, El. J. Combinat. 17 (1) (2010) R29.

FORMULA

It appears that the INVERT transform of factorial numbers A000142 gives 1, 2, 5, 15, 54, 235, 1237, ... - Antti Karttunen, May 30 2003

This is true: translating the defining recurrence to a generating function identity yields A(x)=1/(1-(0!x+1!x^2+2!x^3+...)) which is the INVERT formula.

In other words: let F(x) = sum(n>=0, n!*x^n) then the g.f. is 1/(1-x*F(x)), cf. A052186 (g.f. F(x)/(1+x*F(x))). - Joerg Arndt, Apr 25 2011

a(n) = Sum_{k>=0} A084938(n, k). - Philippe Deléham, Feb 05 2004

G.f. A(x) satisfies: A(x) = (1-x)*A(x)^2 - x^2*A'(x). - Paul D. Hanna, Aug 02 2008

G.f.: A(x) = 1/(1-x/(1-1*x/(1-1*x/(1-2*x/(1-2*x/(1-3*x/(1-3*x...))))))))) (continued fraction). - Paul Barry, Sep 25 2008

From Gary W. Adamson, Jul 22 2011: (Start)

a(n) = upper left term in M^n, M = an infinite square production matrix in which a column of 1's is prepended to Pascal's triangle, as follows:

1, 1, 0, 0, 0,...

1, 1, 1, 0, 0,...

1, 1, 2, 1, 0,...

1, 1, 3, 3, 1,...

... Also, a(n+1) = sum of top row terms of M^n. (End)

G.f.: 1+x/(U(0)-x) where U(k) = 1 + x*k - x*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 10 2012

G.f.: 1/(U(0) - x) where U(k) = 1 - x*(k+1)/(1 - x*(k+1)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 12 2012

a(n) ~ (n-1)! * (1 + 2/n + 7/n^2 + 31/n^3 + 165/n^4 + 1025/n^5 + 7310/n^6 + 59284/n^7 + 543702/n^8 + 5618267/n^9 + 65200918/n^10), for coefficients see A260532. - Vaclav Kotesovec, Jul 28 2015

EXAMPLE

a[ 4 ]=15=a[ 3 ]*0!+a[ 2 ]*1!+a[ 1 ]*2!+a[ 0 ]*3!=5*1+2*1+1*2+1*6.

As to matrix M, a(3) = 5 since the top row of M^n = (5, 5, 4, 1), with a(4) = 15 = (5 + 5 + 4 + 1).

MAPLE

a := proc(n) option remember; `if`(n<2, 1, add(a(n-j-1)*j!, j=0..n-1)) end proc: seq(a(n), n=0..30); # Vaclav Kotesovec, Jul 28 2015

MATHEMATICA

Table[Coefficient[Series[E^x/(E^x-ExpIntegralEi[x]), {x, Infinity, 20}], x, -n], {n, 0, 20}] (* Vaclav Kotesovec, Feb 22 2014 *)

PROG

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x^2*deriv(A)/A)/(1-x)); polcoeff(A, n)} \\ Paul D. Hanna, Aug 02 2008

CROSSREFS

Cf. A051296, A260532.

Row sums of A084938.

Cf. A143965. - Gary W. Adamson, Apr 10 2009

Sequence in context: A193318 A171450 A204190 * A009383 A104429 A109319

Adjacent sequences:  A051292 A051293 A051294 * A051296 A051297 A051298

KEYWORD

easy,nonn

AUTHOR

Leroy Quet

EXTENSIONS

More terms from Vincenzo Librandi, Feb 23 2013

STATUS

approved

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Last modified October 13 18:14 EDT 2019. Contains 327981 sequences. (Running on oeis4.)