%I A026375
%S A026375 1,3,11,45,195,873,3989,18483,86515,408105,1936881,9238023,44241261,
%T A026375 212601015,1024642875,4950790605,23973456915,116312293305,565280386625,
%U A026375 2751474553575,13411044301945,65448142561035,319756851757695
%N A026375 a(n) = number of integer strings s(0),...,s(n) counted by array T in 
               A026374 that have s(n)=0; also a(n)=T(2n,n).
%C A026375 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
%C A026375 Largest coefficient of (1+3*x+x^2)^n; row sums of triangle in A124733 
               . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 02 2007
%C A026375 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
%C A026375 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009: 
               (Start)
%C A026375 Equals INVERT transform of A109033: (1, 2, 6, 22, 88,...), INVERTi transform
%C A026375 of A111966; Binomial transform of A000984, and inverse Binomial transform
%C A026375 of A081671. Convolved with A002212: (1, 3, 10, 36,...) = A026376: (1, 
               6, 30, 144,...).
%C A026375 Equals convolution square root of A003463: (1, 6, 31, 156, 781, 3906,
               ...). (End)
%D A026375 Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%H A026375 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%F A026375 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
%F A026375 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
%F A026375 E.g.f.: exp(3x) I_0(2x), where I_0 is Bessel function. - Michael Somos, 
               Sep 17, 2002
%F A026375 G.f.: 1/sqrt(1-6x+5x^2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Oct 26 2002
%F A026375 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
%F A026375 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
%F A026375 a(n)=A026380(2n-1) (n>0). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Feb 18 2004
%F A026375 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]
%F A026375 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]
%p A026375 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
%o A026375 (PARI) a(n)=if(n<0,0,polcoeff((1+3*x+x^2)^n,n))
%Y A026375 Cf. A085362.
%Y A026375 Cf. A026380.
%Y A026375 First differences are in A085362. Bisection of A026380.
%Y A026375 Cf. A084610. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 18 2009]
%Y A026375 Sequence in context: A151125 A151126 A151127 this_sequence A151128 A049183 
               A049166
%Y A026375 Adjacent sequences: A026372 A026373 A026374 this_sequence A026376 A026377 
               A026378
%K A026375 nonn
%O A026375 0,2
%A A026375 Clark Kimberling (ck6(AT)evansville.edu)