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)}.
The interpretation of T(n,k) as RU walks in terms of M(.,.,.) in the NAME is erroneous. There seems to be a pattern along subdiagonals:
M(3,1,1) = 4 = T(3,2); M(3,1,2) = 1 = T(4,4); M(5,2,1) = 20 = T(5,3); M(5,2,2) = 7 = T(6,5); M(5,2,3) = 1 = T(7,7); M(7,3,0) = 165 = T(6,2); M(7,3,1) = 110 = T(7,4); M(7,3,2) = 44 = T(8,6); M(7,3,3) = 10 = T(9,8); M(7,3,4) = 1 = T(10,10); M(9,4,0) = 1001 = T(8,3); M(9,4,1) = 637 = T(9,5); M(9,4,2) = 273 = T(10,7); M(9,4,3) = 77 = T(11,9); M(9,4,4) = 13 = T(12,11); M(9,4,5) = 1 = T(13,13); M(11,5,0) = 6188 = T(10,4); M(11,5,1) = 3808 = T(11,6); M(11,5,2) = 1700 = T(12,8); M(11,5,3) = 544 = T(13,...); M(11,5,4) = 119; M(11,5,5) = 16; M(11,5,6) = 1; M(13,6,0) = 38760 = T(12,5); M(13,6,1) = 23256 = T(13,7); M(13,6,2) = 10659 = T(14,9); - R. J. Mathar, Jul 31 2024
REFERENCES
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
LINKS
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
FORMULA
For n > 0: Sum_{k>=0} T(n, k) = binomial(2*n-1, n); see A001700. - Philippe Deléham, Feb 13 2004 [Erroneous sum-formula deleted. R. J. Mathar, Jul 31 2024]
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+1); T(n, 2) = Sum_{j>=0} T(n-j, 1)*A000108(j); for k > 2, T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*A000108(j+1). - Philippe Deléham, Feb 13 2004 [Corrected by Sean A. Irvine, Aug 08 2021]
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. - Philippe Deléham, Feb 13 2004
T(n,0) = A033184(n,2). T(n,1) = A033184(n+1,3), T(n,k) = A033184(n+2,k+2) for k>=2. - R. J. Mathar, Jul 31 2024
EXAMPLE
Triangle begins:
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
MAPLE
A050144 := proc(n, k)
if n < k then
0;
elif k =0 then
if n =0 then
0 ;
else
A000108(n-1) ;
end if;
elif k = 1 then
add( procname(n-1-j, 0)*A000108(j+1), j=0..n-1) ;
elif k = 2 then
add( procname(n-j, 1)*A000108(j), j=0..n) ;
else
add( procname(n-1-j, k-1)*A000108(j), j=0..n-1) ;
end if;
end proc:
seq(seq( A050144(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Jul 30 2024
MATHEMATICA
c[n_] := Binomial[2 n, n]/(n + 1);
t[n_, k_] := Which[k == 0, c[n - 1],
k == 1, Sum[t[n - 1 - j, 0]*c[j + 1], {j, 0, n - 2}],
k == 2, Sum[t[n - j, 1]*c[j], {j, 0, n - 1}],
k > 2, Sum[t[n - 1 - j, k - 1] c[j + 1], {j, 0, n - 2}]]
t[0, 0] = 0;
Column[Table[t[n, k], {n, 0, 10}, {k, 0, n}]]
(* Clark Kimberling, Jul 30 2024 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved