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A121462 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having pyramid weight k (1<=k<=n). 5
1, 0, 2, 0, 1, 4, 0, 1, 4, 8, 0, 1, 5, 12, 16, 0, 1, 6, 18, 32, 32, 0, 1, 7, 25, 56, 80, 64, 0, 1, 8, 33, 88, 160, 192, 128, 0, 1, 9, 42, 129, 280, 432, 448, 256, 0, 1, 10, 52, 180, 450, 832, 1120, 1024, 512, 0, 1, 11, 63, 242, 681, 1452, 2352, 2816, 2304, 1024, 0, 1, 12, 75, 316 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

A pyramid in a Dyck word (path) is a factor of the form U^h D^h, where U=(1,1), D=(1,-1) and h is the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.

Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,n)=2^(n-1). Sum(k*T(n,k),k=1..n)=A030267(n)

Mirror image of triangle in A153342 . [From Philippe Deléham, Dec 31 2008]

Essentially triangle given by (0,1/2,1/2,0,0,0,0,0,0,0,...) DELTA (2,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 30 2011

A121462 is jointly generated with A208341 as an array of coefficients of polynomials u(n,x):  initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+x*v(n-1) and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x).  See the Mathematica section. [Clark Kimberling, Mar 11 2012]

LINKS

Table of n, a(n) for n=1..71.

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

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.

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.

FORMULA

T(n,k) = Sum(binomial(k-1,j)*binomial(n-k-1+j,j-1), j=0..k-1) for 2<=k<=n; T(1,1)=1; T(n,1)=0 for n>=2.

G.f.=G=G(t,z)=tz(1-z)/(1-2tz-z+tz^2).

T(n+1,k+1) = A062110(n,k)*2^(2*k-n) . - Philippe Deléham, Aug 01 2006

EXAMPLE

T(4,3)=4 because we have (UD)U(UD)(UD)D, U(UD)(UD)(UD)D, U(UD)(UUDD)D and U(UUDD)(UD)D, where U=(1,1) and D=(1,-1) (the maximal pyramids are shown between parentheses).

Triangle starts:

1;

0,2;

0,1,4;

0,1,4,8;

0,1,5,12,16;

0,1,6,18,32,32;

MAPLE

T:=proc(n, k) if n=1 and k=1 then 1 elif k=1 then 0 elif k<=n then sum(binomial(k-1, j)*binomial(n-k-1+j, j-1), j=0..k-1) else 0 fi end: for n from 1 to 13 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];

v[n_, x_] := x*u[n - 1, x] + (x + 1) v[n - 1, x];

Table[Expand[u[n, x]], {n, 1, z/2}]

Table[Expand[v[n, x]], {n, 1, z/2}]

cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

TableForm[cu]

Flatten[%]    (* A121462 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A208341 *)

(* Clark Kimberling, Mar 11 2012 *)

CROSSREFS

Cf. A001519, A030267, A091866.

Sequence in context: A247489 A208756 A259873 * A271466 A218581 A307177

Adjacent sequences:  A121459 A121460 A121461 * A121463 A121464 A121465

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jul 31 2006

STATUS

approved

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Last modified October 20 15:11 EDT 2019. Contains 328267 sequences. (Running on oeis4.)