%I A085478
%S A085478 1,1,1,1,3,1,1,6,5,1,1,10,15,7,1,1,15,35,28,9,1,1,21,70,84,45,11,1,1,28,
%T A085478 126,210,165,66,13,1,1,36,210,462,495,286,91,15,1,1,45,330,924,1287,
%U A085478 1001,455,120,17,1,1,55,495,1716,3003,3003,1820,680,153,19,1,1,66,715
%N A085478 Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
%C A085478 Coefficient array for Morgan-Voyce polynomial b(n,x). A053122 (unsigned) 
               is the coefficient array for B(n,x). Reversal of A054142. - Paul 
               Barry (pbarry(AT)wit.ie), Jan 19 2004
%C A085478 This triangle is formed from even-numbered rows of triangle A011973 read 
               in reverse order. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 
               16 2004
%C A085478 T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having 
               k+1 peaks. T(n,k) is the number of nondecreasing Dyck paths of semilength 
               n+1, having k peaks at height >=2. T(n,k) is the number of directed 
               column-convex polyominoes of area n+1, having k+1 columns. - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
%C A085478 Riordan array (1/(1-x),x/(1-x)^2). - Paul Barry (pbarry(AT)wit.ie), May 
               09 2005
%C A085478 The triangular matrix a(n,k) = (-1)^(n+k)*T(n,k) is the matrix inverse 
               of A039599 . - Philippe DELEHAM, May 26 2005
%C A085478 The n-th row gives absolute values of coefficients of reciprocal of g.f. 
               of bottom-line of n-wave sequence. - Floor van Lamoen (fvlamoen(AT)planet.nl), 
               Sep 24 2006
%C A085478 Unsigned version of A129818 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 25 2007
%C A085478 T(n, k) is also the number of idempotent order-preserving full transformations 
               (of an n-chain) of height k >=1 (height(alpha) = |Im(alpha)|) and 
               of waist n (waist(alpha) = max(Im(alpha))). [From A. Umar (aumarh(AT)squ.edu.om), 
               Oct 02 2008]
%D A085478 E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck 
               paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
%D A085478 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A085478 E. Deutsch and H. Prodinger, A bijection between directed column-convex 
               polyominoes and ordered trees of height at most three, Theoretical 
               Comp. Science, 307, 2003, 319-325.
%D A085478 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving 
               full transformations. Semigroup Forum 72 (2006), 51-62. [From A. 
               Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%H A085478 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Morgan-VoycePolynomials.html">Morgan-Voyce Polynomials</a>
%F A085478 T(n, k) = (n+k)!/((n-k)!*(2*k)!)
%F A085478 G.f.=(1-z)/[(1-z)^2-tz]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               May 31 2004
%F A085478 Row sums are A001519 (Fib(2n+1)). Diagonal sums are A011782. Binomial 
               transform of A026729 (product of lower triangular matrices). - Paul 
               Barry (pbarry(AT)wit.ie), Jun 21 2004
%F A085478 T(n, 0) = 1, T(n, k) = 0 if n<k; T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) 
               . T(0, 0) = 1, T(0, k) = 0 if k>0; T(n, k) = T(n-1, k-1) + T(n-1, 
               k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) . For the column k, 
               g.f.: Sum_{n>=0} T(n, k)*x^n = (x^k) / (1-x)^(2*k+1) . - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), Feb 15 2004
%F A085478 Sum_{k, 0<=k<=n}T(n,k)*x^(2*k) = A000012(n), A001519(n+1), A001653(n), 
               A078922(n+1), A007805(n), A097835(n), A097315(n), A097838(n), A078988(n), 
               A097841(n), A097727(n), A097843(n), A097730(n), A098244(n), A097733(n), 
               A098247(n), A097736(n), A098250(n), A097739(n), A098253(n), A097742(n), 
               A098256(n), A097767(n), A098259(n), A097770(n), A098262(n), A097773(n), 
               A098292(n), A097776(n) for x=0,1,2,...27,28 respectively . - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Dec 31 2007
%F A085478 T(2n,n)=A005809(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Sep 17 2009]
%p A085478 T:=(n,k)->binomial(n+k,2*k): seq(seq(T(n,k),k=0..n),n=0..11);
%Y A085478 Row sums: A001519. Cf. A007318.
%Y A085478 Cf. A098158 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
%Y A085478 Sequence in context: A102036 A121524 A103141 this_sequence A123970 A129818 
               A055898
%Y A085478 Adjacent sequences: A085475 A085476 A085477 this_sequence A085479 A085480 
               A085481
%K A085478 easy,nonn,tabl
%O A085478 0,5
%A A085478 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 14 2003