%I #44 Jul 31 2024 11:27:43
%S 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,
%T 132,297,572,429,208,65,12,1,429,1001,2002,1638,910,350,90,14,1,1430,
%U 3432,7072,6188,3808,1700,544,119,16,1
%N T(n,k) = M(2n-1,n-1,k-1), 0 <= k <= n, n >= 0, where M(p,q,r) is the number of upright paths from (0,0) to (p,p-q) that meet the line y = x+r and do not rise above it.
%C 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)}.
%C 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:
%C 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
%D 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.
%H Emeric Deutsch and L. Shapiro, <a href="https://doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%H R. K. Guy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%F 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]
%F 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]
%F 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
%F T(n,0) = A033814(n,2). T(n,1) = A033814(n+1,3), T(n,k) = A033814(n+2,k+2) for k>=2. - _R. J. Mathar_, Jul 31 2024
%e 0
%e 1 0
%e 1 1 1
%e 2 3 4 1
%e 5 9 14 5 1
%e 14 28 48 20 6 1
%e 42 90 165 75 27 7 1
%e 132 297 572 275 110 35 8 1
%e 429 1001 2002 1001 429 154 44 9 1
%e 1430 3432 7072 3640 1638 637 208 54 10 1
%e 4862 11934 25194 13260 6188 2548 910 273 65 11 1
%p A050144 := proc(n,k)
%p if n < k then
%p 0;
%p elif k =0 then
%p if n =0 then
%p 0 ;
%p else
%p A000108(n-1) ;
%p end if;
%p elif k = 1 then
%p add( procname(n-1-j,0)*A000108(j+1),j=0..n-1) ;
%p elif k = 2 then
%p add( procname(n-j,1)*A000108(j),j=0..n) ;
%p else
%p add( procname(n-1-j,k-1)*A000108(j),j=0..n-1) ;
%p end if;
%p end proc:
%p seq(seq( A050144(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jul 30 2024
%t c[n_] := Binomial[2 n, n]/(n + 1);
%t t[n_, k_] := Which[k == 0, c[n - 1],
%t k == 1, Sum[t[n - 1 - j, 0]*c[j + 1], {j, 0, n - 2}],
%t k == 2, Sum[t[n - j, 1]*c[j], {j, 0, n - 1}],
%t k > 2, Sum[t[n - 1 - j, k - 1] c[j + 1], {j, 0, n - 2}]]
%t t[0, 0] = 0;
%t Column[Table[t[n, k], {n, 0, 10}, {k, 0, n}]]
%t (* _Clark Kimberling_ July 30 2024 *)
%Y {M(2n, 0, k)} is given by A039599. {M(2n+1, n+1, k+1)} is given by A039598.
%Y Cf. A033184, A050153, A000108 (column 0), A000245 (column 1), A002057 (column 2), A000344 (column 3), A003517 (column 4), A000588 (column 5), A003518 (column 6), A001392 (column 7), A003519 (column 8), A000589 (column 9), A090749 (column 10).
%K nonn,tabl
%O 0,7
%A _Clark Kimberling_