%I A026378
%S A026378 1,4,17,75,339,1558,7247,34016,160795,764388,3650571,17501619,
%T A026378 84179877,406020930,1963073865,9511333155,46169418195,224484046660,
%U A026378 1093097083475,5329784874185,26018549129545,127154354598330,622031993807565
%N A026378 a(n) = number of integer strings s(0),...,s(n) counted by array T in 
               A026374 that have s(n)=1; also a(n)=T(2n-1,n-1).
%C A026378 Number of lattice paths from (0,0) to the line x=n-1 that do not go below 
               the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types 
               of steps H=(1,0) (left factors of 3-Motzkin steps). Example: a(3)=17 
               because we have UD, UU, 9 HH paths, 3 HU paths and 3 UH paths. - 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 22 2004
%C A026378 Also a(n) = number of integer strings s(0), ..., s(n) counted by array 
               U in A026386 that have s(n)=1; a(n) = U(2n-1, n-1).
%C A026378 The Hankel transform of [1,1,4,17,75,339,1558,...] is [1,3,8,21,55,144,
               377,...] (see A001906) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Apr 13 2007
%C A026378 Number of peaks in all skew Dyck paths of semilength n. A skew Dyck path 
               is a path in the first quadrant which begins at the origin, ends 
               on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and 
               L=(-1,-1)(left) so that up and left steps do not overlap. The length 
               of the path is defined to be the number of its steps. Example: a(2)=4 
               because in the 3 (=A002212(2)) skew Dyck paths (UD)(UD), U(UD)D and 
               U(UD)L we have altogether 4 peaks (shown between parentheses). - 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 25 2007
%C A026378 Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,...] . 
               - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
%C A026378 5th binomial transform of (-1)^n*A000108. [From Paul Barry (pbarry(AT)wit.ie), 
               Jan 13 2009]
%C A026378 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2009: 
               (Start)
%C A026378 Convolved with A007317, (1, 2, 5, 15, 51,...) = A026376: (1, 6, 30, 144,
               ...)
%C A026378 Equals A026375, (1, 3, 11, 45, 195,...) convolved with A002212 prefaced 
               with
%C A026378 a 1: (1, 1, 3, 10, 36, 137,...). (End)
%D A026378 E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).
%H A026378 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 A026378 G.f.: (1/2)/(5*x^2-x)*(1-5*x-(1-6*x+5*x^2)^(1/2)). E.g.f.: exp(3*x)*(BesselI(0, 
               2*x)+BesselI(1, 2*x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 
               03 2003
%F A026378 G.f.= [(1-z)/sqrt(1-6z+5z^2)-1]/2 = z + 4z^2 + 17z^3 + ... - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Jan 22 2004
%F A026378 a(n) = coefficient of t^n in (1+t)(1+3t+t^2)^(n-1). - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Jan 30 2004
%F A026378 a(n)=A026380(2n-2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 
               18 2004
%F A026378 a(n)=[2(3n-2)a(n-1)-5(n-2)a(n-2)]/n for n>=2; a(0)=0, a(1)=1. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Mar 18 2004
%F A026378 a(n+1) = sum(k=0, n, binomial(n, k)*sum(i=0, k, binomial(k+i, i))) - 
               Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2004
%F A026378 a(n+1) = sum(k=0, n, binomial(n, k)*binomial(2*k+1, k+1)) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Aug 06 2004
%F A026378 a(n)=Sum(k*A126182(n-1,k-1),k=1..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jul 25 2007
%F A026378 Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 13 2009: (Start)
%F A026378 G.f.: (1/(1-5x))*c(-x/(1-5x)), c(x) the g.f. of A000108;
%F A026378 a(n)=sum{k=0..n, C(n,k)*(-1)^k*A000108(k)*5^(n-k)} (offset 0). (End)
%Y A026378 Cf. A002212, A026375.
%Y A026378 Cf. A026380.
%Y A026378 Half the values of A026387. Bisection of A026380 and A026392.
%Y A026378 Cf. A126182.
%Y A026378 A026375, A026376, A007317, A002212 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 17 2009]
%Y A026378 Sequence in context: A049027 A026751 A081568 this_sequence A151247 A117439 
               A081910
%Y A026378 Adjacent sequences: A026375 A026376 A026377 this_sequence A026379 A026380 
               A026381
%K A026378 nonn
%O A026378 1,2
%A A026378 Clark Kimberling (ck6(AT)evansville.edu)