0,3

Comment from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 25 2004: "Let B = 1/sqrt(1-4z) = g.f. for central binomial coeffs (A000984); F = (1-sqrt(1-4z))/[z(3-sqrt(1-4z))] = g.f. for (A000957).

"B = 1 + 2z + 6z^2 + 20z^3 + ... gives the number of nodes in all ordered trees with 0,1,2,3,... edges. On p. 288 of the Deutsch-Shapiro paper one finds that zBF=z + 2z^2 + 7z^3 + 24z^4 + ... gives the number of nodes of odd outdegree in all ordered trees with 1,2,3,... edges (cf. A014300).

"Consequently, B - zBF = 2/[3sqrt(1-4z)-1+4z] = 1 + z + 4z^2 + 13z^3 + 46z^4 + ... gives the total number of nodes of even degree in all ordered trees with 0,1,2,3,4,... edges."

Main diagonal of the following array: first column is filled with 1's, first row is filled alternatively with 1's or 0's: m(i,j)=m(i-1,j)+m(i,j-1): 1 0 1 0 1 ... / 1 1 2 2 3 ... / 1 2 4 6 9 ... / 1 3 7 13 22 ... / 1 4 11 24 46 ... - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 05 2002

The Hankel transform of [1,1,4,13,46,166,610,2269,...] is 3^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 08 2007

Second binomial transform of A127361 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 14 2007

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

G.f. is logarithmic derivative of the generating function for the Catalan numbers A000108. So this sequence might be called the "log-Catalan" numbers. - Murray R Bremner (bremner(AT)iastate.edu) Jan 25 2004

Sum('binomial(k+n, n-k)*binomial(n-k, k)', 'k'=0..floor(n/2)) - Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001

G.f.: 2/(3sqrt(1-4z)-1+4z) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 09 2002

a(n) = (-1)^n*sum(k=0, n, (-1)^k*C(n+k, k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 20 2002

a(n)=sum(binomial(2n-2j-1, n-1), j=0..floor(n/2)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 28 2004

A Catalan transform of the Jacobsthal numbers A001045(n+1) under the mapping G(x)-> G(xc(x)), c(x) the g.f. of A001008. The inverse mapping is H(x)->H(x(1-x)). a(n)=sum{k=0..n, (k/(2n-k))binomial(2n-k, n-k)A001045(k+1)} - Paul Barry (pbarry(AT)wit.ie), Dec 18 2004

a(n)=sum{k=0..n, binomial(2n-k, k)*binomial(k, n-k)}; - Paul Barry (pbarry(AT)wit.ie), Jul 25 2005

a(n)=Sum[k, 0<=k<=n-1}A126093(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 08 2007

a(n)=(-1/2)^(n+2)+(2/3)*Sum([(4^n-k)*(binomial(2k,k))*(1/(1-2*k))*(1-(-1/8)^(n-k+1))],k=0..n) and a(n)=(-1/2)^(n+2)+(3/4)*Sum(((-1/2)^(n-k))*(binomial(2k,k)),k=0..n) - Aktar Yalcin (aktaryalcin(AT)msn.com), Jul 06 2007

seq(add((binomial(k+n, n-k)*binomial(n-k, k)), k=0..floor(n/2)), n=1..30);

(PARI) a(n)=(-1)^n*sum(k=0, n, (-1)^k*binomial(n+k, k))

Adjacent sequences: A026638 A026639 A026640 this_sequence A026642 A026643 A026644

Sequence in context: A104460 A095128 A047154 this_sequence A087440 A014145 A098841

nonn

Clark Kimberling (ck6(AT)evansville.edu)