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A101371 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges and k leaves at level 1.

%I A101371 #28 Jul 21 2024 16:10:08
%S A101371 1,0,1,2,0,1,7,4,0,1,34,14,6,0,1,171,72,21,8,0,1,905,370,114,28,10,0,
%T A101371 1,4952,1995,597,160,35,12,0,1,27802,11064,3278,852,210,42,14,0,1,
%U A101371 159254,62774,18420,4762,1135,264,49,16,0,1,927081,362614,105618,27104,6455,1446,322,56,18,0,1
%N A101371 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges and k leaves at level 1.
%C A101371 Row n has n+1 terms.
%H A101371 Andrew Howroyd, <a href="/A101371/b101371.txt">Table of n, a(n) for n = 0..1274</a>
%H A101371 Naiomi Cameron and J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
%H A101371 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of non-crossing configurations</a>, Discrete Math., 204, 203-229, 1999.
%H A101371 M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00121-0">Enumeration of noncrossing trees on a circle</a>, Discrete Math., 180, 301-313, 1998.
%F A101371 T(n, k) = Sum_{i=0..n-k} (-1)^i*((k+i+1)/(2n-k-i+1)) binomial(k+i, i) binomial(3n-2k-2i, n-k-i) for 0 <= k <= n.
%F A101371 G.f.: g/(1+z*g-t*z*g), where g = 1+z*g^3.
%e A101371 Triangle begins:
%e A101371     1;
%e A101371     0,  1;
%e A101371     2,  0,  1;
%e A101371     7,  4,  0, 1;
%e A101371    34, 14,  6, 0, 1;
%e A101371   171, 72, 21, 8, 0, 1;
%e A101371   ...
%p A101371 T:=proc(n,k) if k<=n then sum((-1)^i*(k+i+1)*binomial(k+i,i)*binomial(3*n-2*k-2*i,n-k-i)/(2*n-k-i+1),i=0..n-k) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t A101371 t[n_, k_] := Sum[(-1)^i*(k + i + 1)/(2n - k - i + 1)*Binomial[k + i, i]* Binomial[3n - 2k - 2i, n - k - i], {i, 0, n - k}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 21 2013, after Maple *)
%o A101371 (PARI) T(n, k) = sum(i=0, n-k, (-1)^i*(k+i+1)*binomial(k+i, i)*binomial(3*n-2*k-2*i, n-k-i)/(2*n-k-i+1)); \\ _Andrew Howroyd_, Nov 06 2017
%Y A101371 Columns k=0..2 give A023053, A374835, A374836.
%Y A101371 Row sums give A001764.
%Y A101371 Cf. A065600.
%K A101371 nonn,tabl
%O A101371 0,4
%A A101371 _Emeric Deutsch_, Jan 14 2005