cp's OEIS Frontend

A050144 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.

%I A050144 #53 Aug 03 2025 00:06:08
%S A050144 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 A050144 132,297,572,429,208,65,12,1,429,1001,2002,1638,910,350,90,14,1,1430,
%U A050144 3432,7072,6188,3808,1700,544,119,16,1
%N A050144 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 A050144 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 A050144 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 A050144  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 A050144 Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%H A050144 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 A050144 Boris A. Bondarenko, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a>, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
%H A050144 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 A050144 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 A050144 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 A050144 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 A050144 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
%e A050144 Triangle begins:
%e A050144      0
%e A050144      1    0
%e A050144      1    1    1
%e A050144      2    3    4    1
%e A050144      5    9   14    6    1
%e A050144     14   28   48   27    8    1
%e A050144     42   90  165  110   44   10    1
%e A050144    132  297  572  429  208   65   12    1
%e A050144    429 1001 2002 1638  910  350   90   14    1
%e A050144   1430 3432 7072 6188 3808 1700  544  119   16    1
%p A050144 A050144 := proc(n,k)
%p A050144     if n < k then
%p A050144         0;
%p A050144     elif k =0 then
%p A050144         if n =0 then
%p A050144             0 ;
%p A050144         else
%p A050144             A000108(n-1) ;
%p A050144         end if;
%p A050144     elif k = 1 then
%p A050144         add( procname(n-1-j,0)*A000108(j+1),j=0..n-1) ;
%p A050144     elif k = 2 then
%p A050144         add( procname(n-j,1)*A000108(j),j=0..n) ;
%p A050144     else
%p A050144         add( procname(n-1-j,k-1)*A000108(j),j=0..n-1) ;
%p A050144     end if;
%p A050144 end proc:
%p A050144 seq(seq( A050144(n,k),k=0..n),n=0..12) ; # _R. J. Mathar_, Jul 30 2024
%t A050144 c[n_] := Binomial[2 n, n]/(n + 1);
%t A050144 t[n_, k_] := Which[k == 0, c[n - 1],
%t A050144   k == 1, Sum[t[n - 1 - j, 0]*c[j + 1], {j, 0, n - 2}],
%t A050144   k == 2, Sum[t[n - j, 1]*c[j], {j, 0, n - 1}],
%t A050144   k > 2, Sum[t[n - 1 - j, k - 1] c[j + 1], {j, 0, n - 2}]]
%t A050144 t[0, 0] = 0;
%t A050144 Column[Table[t[n, k], {n, 0, 10}, {k, 0, n}]]
%t A050144 (* _Clark Kimberling_, Jul 30 2024 *)
%Y A050144 {M(2n, 0, k)} is given by A039599. {M(2n+1, n+1, k+1)} is given by A039598.
%Y A050144 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 A050144 nonn,tabl
%O A050144 0,7
%A A050144 _Clark Kimberling_