1,4

Comment from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 24 2005 (Start): Let A be the array:

1, 0, 0, 0, 0, 0,...

0, 1, 0, 0, 0, 0,...

1, 0, 1, 0, 0, 0,...

0, 2, 0, 1, 0, 0,...

1, 0, 3, 0, 1, 0,...

0, 3, 0, 4, 0, 1,...

...

where columns are bin(n,k) with alternating zeros. (Row sums = 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...(Fibonacci numbers)) Let P = infinite lower triangular Pascal triangle matrix (A007318). Form P * A: this gives the rows of the present sequence.. (End)

T(n,k) is the number of nondecreasing Dyck paths of semilength n, having height of rightmost peak equal to k. Example: T(4,1)=4 because we have UDUDUDUD, UDUUDDUD, UUDDUDUD and UUUDDDUD, where U=(1,1) and D=(1,-1). Sum of row n = fibonacci(2n-1) (A001519). Basically the same as A062110.

T(n,k) is the number of permutations of [n] with length n-k that avoid the patterns 321 and 3412. - Bridget Eileen Tenner (bridget(AT)math.mit.edu), Sep 28 2005

T(2*n-1,n)/n = A001003(n-1) (little Schroeder numbers). Proof with Lagrange inversion of inverse of g.f. of A001003.

Row sums = odd indexed Fibonacci numbers.

Diagonal sums : A077998 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 16 2008]

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

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.

V. E. Hoggatt, Jr. and Marjorie Bicknell, editors: "A Primer for the Fibonacci Numbers", 1970, p. 87

T(n, k)=sum(binomial(k+j, k-1)*binomial(n-k-1, j), j=0..n-k-1) (0<=k<=n). G.f.=tz(1-z)/[1-2z-tz(1-z)].

Triangle begins:

1;

1, 1;

2, 2, 1;

4, 5, 3, 1;

8, 12, 9, 4, 1;

16, 28 25, 14, 5, 1;

32, 64, 66, 44, 20, 6, 1;

64, 144, 168, 129, 70, 27, 7, 1;

...

T:=proc(n, k) if k<n-1 then sum(binomial(k+j, k-1)*binomial(n-k-1, j), j=0..n-k-1) elif k=n-1 then n-1 elif k=n then 1 else 0 fi end: for n from 1 to 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

Cf. A001519. Essentially the same array as A062110.

Row sums = A001519(n-1), n>=1.

Sequence in context: A145036 A001404 A104580 this_sequence A064189 A063415 A098977

Adjacent sequences: A105303 A105304 A105305 this_sequence A105307 A105308 A105309

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 25 2005

Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Apr 27 2007

Corrected comment. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 09 2008