0,3

Number of ordered partitions of n into n parts, allowing zeros (cf. A097070) is binomial(2*n-1,n) = a(n) = essentially A001700. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 15 2004

a(n) = A110556(n)*(-1)^n, central terms in triangle A110555. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005

Hankel transform is A000027; example: Det([1,1,3,10;1,3,10,35;3,10,35,126;10,35,126,462])=4 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 13 2007

a(n) is the number of functions f:[n]->[n] such that for all x,y in [n] if x<y then f(x)<=f(y). So 2*a(n)-n=A045992(n) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 02 2009]

Hankel transform of the aeration of this sequence is A000027 doubled: 1,1,2,2,3,3,... [From Paul Barry (pbarry(AT)wit.ie), Sep 26 2009]

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

a(n)=(0^n+C(2n, n))/2. - Paul Barry (pbarry(AT)wit.ie), May 21 2004

a(n) is the coefficient of x^n in 1/(1-x)^n and also the sum of the first n coefficients of 1/(1-x)^n. Given B(x) with the property that the coefficient of x^n in B(x)^n equals the sum of the first n coefficients of B(x)^n, then B(x)=B(0)/(1-x).

G.f.: 1/(2-C(x)) where C(x) is g.f. for Catalan numbers A000108.

G.f.: (1+1/sqrt(1-4x))/2. a(n)=binomial(2n-1,n).

a(n)=sum{k=0..n, binomial(2n, k)cos((n-k)*pi)}; a(n)=sum{k=0..n, binomial(n, (n-k)/2)(1+(-1)^(n-k))cos(k*pi/2)/2} (with interpolated zeros); a(n)=sum{k=0..floor(n/2), binomial(n, k)cos((n-2k)pi/2)} (with interpolated zeros); - Paul Barry (pbarry(AT)wit.ie), Nov 02 2004

a(n)=Sum_{k, 0<=k<=n}A094527(n,k)*(-1)^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 14 2007

The five rooted ordered trees with 3 edges have 10 leaves.

..x........................

..o..x.x..x......x.........

..o...o...o.x..x.o..x.x.x..

..r...r....r....r.....r....

seq(abs(binomial(-n, -2*n)), n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007

(PARI) a(n)=sum(i=0, n, binomial(n+i-2, i))

(PARI) a(n)=if(n<0, 0, polcoeff((1+1/sqrt(1-4*x+x*O(x^n)))/2, n))

(PARI) a(n)=if(n<0, 0, polcoeff(1/(1-x+x*O(x^n))^n, n))

(PARI) a(n)=if(n<0, 0, binomial(2*n-1, n))

(PARI) {a(n)=if(n<1, n==0, polcoeff( subst((1-x)/(1-2*x), x, serreverse(x-x^2+x*O(x^n))), n))}

A001700(n)=a(n+1). a(n)=A024718(n)-A024718(n-1).

Sequence in context: A099908 A167403 A001700 this_sequence A110556 A072266 A085282

Adjacent sequences: A088215 A088216 A088217 this_sequence A088219 A088220 A088221

nonn

Michael Somos, Sep 24 2003

Essentially the same as A001700, which has much more information.