# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a318942 Showing 1-1 of 1 %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). Enumerations of peaks and valleys on non-decreasing Dyck paths. 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 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE