0,5

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

This triangle is formed from even-numbered rows of triangle A011973 read in reverse order. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004

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

Riordan array (1/(1-x),x/(1-x)^2). - Paul Barry (pbarry(AT)wit.ie), May 09 2005

The triangular matrix a(n,k) = (-1)^(n+k)*T(n,k) is the matrix inverse of A039599 . - Philippe DELEHAM, May 26 2005

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

Unsigned version of A129818 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 25 2007

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]

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.

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 H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.

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]

Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

T(n, k) = (n+k)!/((n-k)!*(2*k)!)

G.f.=(1-z)/[(1-z)^2-tz]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004

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

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

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

T(2n,n)=A005809(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]

T:=(n, k)->binomial(n+k, 2*k): seq(seq(T(n, k), k=0..n), n=0..11);

Row sums: A001519. Cf. A007318.

Cf. A098158 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]

Sequence in context: A102036 A121524 A103141 this_sequence A123970 A129818 A055898

Adjacent sequences: A085475 A085476 A085477 this_sequence A085479 A085480 A085481

easy,nonn,tabl

DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 14 2003