0,4

Number of Dyck paths of semilength n having exactly one long ascent (i.e. ascent of length at least two). Example: a(4)=11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD, and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1). Also number of ordered trees with n edges having exactly one branch node (i.e. vertex of outdegree at least two). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2004

Number of permutations of {1,2,...,n} with exactly one descent (i.e. permutations (p(1),p(2),...,p(n)) such that #{i: p(i)>p(i+1)}=1). E.g. a(3)=4 because the permutations of {1,2,3} with one descent are 132, 213, 231 and 312.

A107907(a(n+2)) = A000079(n+2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), May 28 2005

a(n+1) is the convolution of nonnegative integers (A001477) and powers of two (A000079) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006

Number of partitions of an n-set having exactly one block of size > 1. Example: a(4)=11 because, if the partitioned set is {1,2,3,4}, then we have 1234, 123|4, 124|3, 134|2, 1|234, 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3, and 1|2|34. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 28 2006

n divides a(n+1) for n = A014741(n) = {1, 2, 6, 18, 42, 54, 126, 162, 294, 342, 378, 486, 882, 1026, ...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 03 2006

(Number of permutations avoiding patterns 321, 2413, 3412, 21534) minus one. - J.-L. Baril (barjl(AT)u-bourgogne.fr), Nov 01 2007, Mar 21 2008

O. Bottema, Problem #562, Nieuw Archief voor Wiskunde, 28 (1980) 115.

F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.

Pascal Floquet, Serge Domenech and Luc Pibouleau, "Combinatorics of Sharp Separation System synthesis : Generating functions and Search Efficiency Criterion", Industrial Engineering and Chemistry Research, 33, pp. 440-443, 1994

Pascal Floquet, Serge Domenech, Luc Pibouleau and Said Aly, "Some Complements in Combinatorics of Sharp Separation System Synthesis", American Institute of Chemical Engineering Journal, 39(6), pp. 975-978, 1993.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 34.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 20 (1968), 8-16.

T. D. Noe, Table of n, a(n) for n=0..500

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 388

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

G.f.: x^2/((1-2*x)*(1-x)^2). a(1)=0, a(n)=2*a(n-1)+n-1.

a(n)=sum{k=2..n, binomial(n, k) } - Paul Barry (pbarry(AT)wit.ie), Jun 05 2003

a(n+1)=sum(1<=i<=n, sum(1<=j<=i, C(i, j))) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 07 2003

a(n+1)=2^n*sum(k=0, n, k/2^k) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 26 2003

a(0)=0, a(1)=0, a(n) = Sum i=0..n-1 i+a(i) for i > 1 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 12 2004

a(n+1)=sum{k=0..n, (n-k)2^k}=sum{k=0..n, k*2^(n-k)} - Paul Barry (pbarry(AT)wit.ie), Jul 29 2004

a(n)=sum{k=0..n, binomial(n, k+2)}; a(n+2)=sum{k=0..n, binomial(n+2, k+2)}. - Paul Barry (pbarry(AT)wit.ie), Aug 23 2004

a(n)=sum{k=0..floor((n-1)/2), binomial(n-k-1, k+1)2^(n-k-2)*(-1/2)^k} - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004

a(0)=0, a(1)=0, a(n)=3*a(n-1)-2*a(n-2)+1 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005

G.f.=exp(x)[exp(x)-1-x]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 28 2006

a(0) = 0; a(n)=stirling2(n,2)+a(n-1). - Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 18 2007

Equals row sums of triangle A130128 starting (1, 4, 11, 26, 57,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 11 2007

Starting (1, 4, 11, 26,...) gives row sums of triangle A130330 which is composed of (1,3,7,15...) in every column, thus: row sums of (1; 3,1; 7,3,1;...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 24 2007

a(n) = 2*a(n-1)+n-1. - Graeme McRae (g_m(AT)mcraefamily.com), Jun 15 2007

Row sums of triangle A131768 starting (1, 4, 11, 26, 57, 120,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 13 2007

Sequence starting (1, 4, 11, 26, 57,...) = A130321 * (1, 2, 3,...). Sequence starting (1, 4, 11, 26, 57,...) = binomial transform of (1, 3, 4, 4, 4,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2007

Row sums of triangle A131952 starting (1, 4, 11, 26,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 30 2007

[ seq(2^n-n-1, n=1..50) ];

a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=3*a[n-1]-2*a[n-2]+1 od: seq(a[n], n=0..50); (Kristof)

seq(add(binomial(n, k)*(bell(k-n)), k=2..n), n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006

a:=n->sum (2^j-1, j=1..n): seq(a(n), n=-1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2007

with(combinat): a:=n->(sum((stirling2(j, 2)), j=0..n)): seq(a(n), n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007

A000295:=-z/(2*z-1)/(z-1)**2; [S. Plouffe in his 1992 dissertation.]

a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=n+2*a[n-1]-1 od: seq(a[n], n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008

Cf. A008949, A000079, A000225, A002663, A002664, A035039-A035042, A008292 (the main entry for the Eulerian numbers).

Cf. A000108, A014741, A130128, A130330, A131768, A130321, A131952.

Partial sums of A000225.

Row sums of triangle A014473. Second column of triangles A112493 and A112500.

Adjacent sequences: A000292 A000293 A000294 this_sequence A000296 A000297 A000298

Sequence in context: A027660 A002940 A030196 this_sequence A125128 A130103 A034334

nonn,easy,nice

njas