0,3

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 (malbert(AT)cs.otago.ac.nz), Dec 14 2004

Starting (1, 2, 5, 15,...) = row sums of triangle A143965 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]

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

Leroy Quet, Home Page (listed in lieu of email address)

David Callan, A combinatorial interpretation of the eigensequence for composition

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

a(n) = Sum_{k>=0} A084938(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 05 2004

Regarding Karttunen's conjecture: 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.

G.f. A(x) satisfies: A(x) = (1-x)*A(x)^2 - x^2*A'(x). [From Paul D. Hanna (pauldhanna(AT)juno.com), 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); [From Paul Barry (pbarry(AT)wit.ie), Sep 25 2008]

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

(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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Aug 02 2008]

Cf. A051296.

Row sums of A084938.

A143965 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]

Sequence in context: A006966 A056841 A107112 this_sequence A009383 A104429 A109319

Adjacent sequences: A051292 A051293 A051294 this_sequence A051296 A051297 A051298

easy,nonn

Leroy Quet