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A318942 Triangle read by rows: T(n,k) = number of Dyck paths with n nodes and altitude k (1 <= k <= n).

%I A318942 #27 Sep 25 2022 11:28:35
%S A318942 1,2,1,5,4,1,13,12,6,1,34,35,21,8,1,89,99,68,32,10,1,233,274,208,114,
%T A318942 45,12,1,610,747,612,376,175,60,14,1,1597,2015,1752,1177,620,253,77,
%U A318942 16,1,4181,5394,4916,3549,2062,959,350,96,18,1,10946,14359,13588,10406,6551,3381,1414,468,117,20
%N A318942 Triangle read by rows: T(n,k) = number of Dyck paths with n nodes and altitude k (1 <= k <= n).
%H A318942 Czabarka, É., Flórez, R., Junes, L., & Ramírez, J. L. (2018). <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>. Discrete Mathematics, 341(10), 2789-2807.
%F A318942 Czabarka et al. give a g.f. - _N. J. A. Sloane_, Apr 09 2019
%e A318942 Triangle begins:
%e A318942 1,
%e A318942 2,1,
%e A318942 5,4,1,
%e A318942 13,12,6,1,
%e A318942 34,35,21,8,1,
%e A318942 89,99,68,32,10,1,
%e A318942 233,274,208,114,45,12,1,
%e A318942 610,747,612,376,175,60,14,1,
%e A318942 1597,2015,1752,1177,620,253,77,16,1,
%e A318942 ...
%p A318942 A318942 := proc(n,k) # Theorem 7 of Czabarka et al.
%p A318942     option remember;
%p A318942     if k = 1 then
%p A318942         combinat[fibonacci](2*n-1) ;
%p A318942     elif n =k then
%p A318942         1;
%p A318942     elif n = k+1 then
%p A318942         2*procname(n-1,k)+procname(n-1,k-1) ;
%p A318942     elif n >= k+2 then
%p A318942         2*procname(n-1,k)+procname(n-1,k-1)-procname(n-2,k-1)+combinat[fibonacci](2*n-2*k-2)  ;
%p A318942     else
%p A318942         0 ;
%p A318942     end if;
%p A318942 end proc:
%p A318942 seq( seq(A318942(n,k),k=1..n),n=1..12 ) ; # _R. J. Mathar_, Apr 09 2019
%t A318942 T[n_, k_] := T[n, k] = Which[k == 1, Fibonacci[2*n - 1], n == k, 1, n == k + 1, 2*T[n - 1, k] + T[n - 1, k - 1], n >= k + 2, 2*T[n - 1, k] + T[n - 1, k - 1] - T[n - 2, k - 1] + Fibonacci[2*n - 2*k - 2], True, 0];
%t A318942 Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Sep 25 2022, after _R. J. Mathar_ *)
%Y A318942 Col. 1 is alternate Fibonaccis, cols. 2, 3, 4 are A318941, A318943, A318944.
%Y A318942 Row sums give A038731(n-1).
%K A318942 nonn,tabl,easy
%O A318942 1,2
%A A318942 _N. J. A. Sloane_, Sep 18 2018