0,2

Partial sums of A085362. Number of bilateral Schroeder paths (i.e. lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at odd (positive or negative) levels. Example: a(2)=11 because we have HUD, UDH, UDUD, UUDD, UDDU, their reflections in the x-axis and HH. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 30 2004

Largest coefficient of (1+3*x+x^2)^n; row sums of triangle in A124733 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 02 2007

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in three colors. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Feb 05 2008

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009: (Start)

Equals INVERT transform of A109033: (1, 2, 6, 22, 88,...), INVERTi transform

of A111966; Binomial transform of A000984, and inverse Binomial transform

of A081671. Convolved with A002212: (1, 3, 10, 36,...) = A026376: (1, 6, 30, 144,...).

Equals convolution square root of A003463: (1, 6, 31, 156, 781, 3906,...). (End)

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Representation by Gauss's hypergeometric function, in Maple notation: a(n)=hypergeom([ -n, 1/2 ], [ 1 ], -4). - Karol A. Penson (penson(AT)lptl.jussieu.fr), Apr 20 2001

This sequence is the binomial transform of A000984 - Johh W. Layman (layman(AT)math.vt.edu), Aug 11 2000; proved by Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 26 2002

E.g.f.: exp(3x) I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 17, 2002

G.f.: 1/sqrt(1-6x+5x^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 26 2002

na(n)-3(2n-1)a(n-1)+5(n-1)a(n-2)=0 for n > 1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 24 2004

a(n)=[t^n](1+3t+t^2)^n; a(n)=sum(3^(2j-n)*binomial(n, j)*binomial(j, n-j), j=ceil(n/2)..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 30 2004

a(n)=A026380(2n-1) (n>0). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 18 2004

G.f.: 1/(1-x-2x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x... (continued fraction); [From Paul Barry (pbarry(AT)wit.ie), Jan 06 2009]

a(n) = sum of squared coeficients of (1+x-x^2)^n - see triangle A084610. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 18 2009]

sum('binomial(n, k)*binomial(2*k, k)', 'k'=0..n); seq( sum('binomial(n, k)*binomial(2*k, k)', 'k' =0..floor(n)), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001

(PARI) a(n)=if(n<0, 0, polcoeff((1+3*x+x^2)^n, n))

Cf. A085362.

Cf. A026380.

First differences are in A085362. Bisection of A026380.

Cf. A084610. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 18 2009]

Sequence in context: A151125 A151126 A151127 this_sequence A151128 A049183 A049166

Adjacent sequences: A026372 A026373 A026374 this_sequence A026376 A026377 A026378

nonn

Clark Kimberling (ck6(AT)evansville.edu)