|
| |
|
|
A050144
|
|
T(n,k)=M(2n-1,n-1,k-1), 0<=k<=n, n >= 0, where M(p,q,r)=number of upright paths from (0,0) to (p,p-q) that meet the line y=x-r and do not rise above it.
|
|
9
| |
|
|
0, 1, 0, 1, 1, 1, 2, 3, 4, 1, 5, 9, 14, 6, 1, 14, 28, 48, 27, 8, 1, 42, 90, 165, 110, 44, 10, 1, 132, 297, 572, 429, 208, 65, 12, 1, 429, 1001, 2002, 1638, 910, 350, 90, 14, 1, 1430, 3432, 7072, 6188, 3808, 1700, 544, 119, 16, 1
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,7
|
|
|
COMMENTS
| Let V=(e(1),...,e(n)) consist of q 1's and p-q 0's; let V(h)=(e(1),...,e(h)) and m(h)=(#1's in V(h))-(#0's in V(h)) for h=1,...,n. Then M(p,q,r)=number of V having r=max{m(h)}.
First 6 columns of T are A000108, A000245, A002057, A003517, A003518, A003519.
|
|
|
REFERENCES
| B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
|
|
|
LINKS
| R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
|
|
|
FORMULA
| For n>0: Sum_{k>=0} T(n, k)= binomial(2*n-1, n); see A001700. Sum_{k>=1} (-1)^k*T(n, k) = 0 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 13 2004
T(n, k) = 0 if n<k; T(0, 0)= 0, T(n, 0)= A000108(n-1) for n>0; T(n, 1)= sum_{j>=0} T(n-1-j, 0)*A000108(j); T(n, 2)= sum_{j>=0} T(n-j, 1)*A000108(j+1); for k>2, T(n, k)= sum_{j>=0} T(n-1-j, k-1)*A000108(j+1) . For the column k=0, G.f.: x*C(x); for the column k=1, G.f.: x*C(x)*(C(x)-1); for the column k, k>1, G.f.: x*C(x)^2*(C(x)-1)^(k-1); where C(x)= sum_{n>=0} A000108(n)*x^n is G.f. for Catalan numbers, A000108 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 13 2004
|
|
|
EXAMPLE
| Rows: {0}; {1,0}; {1,1,1}; ...
|
|
|
CROSSREFS
| {M(2n, n, k)} is given by A039599. {M(2n+1, n+1, k+1)} is given by A039598.
Sequence in context: A067003 A117386 A101174 * A124406 A065331 A066262
Adjacent sequences: A050141 A050142 A050143 * A050145 A050146 A050147
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
|
| |
|
|
