cp's OEIS Frontend

A348013 Triangle by rows: T(n,k) is the number of n-step Dyck paths with k catastrophes.

%I A348013 #9 Sep 25 2021 06:46:54
%S A348013 1,1,1,3,2,1,4,7,3,1,10,14,12,4,1,15,37,31,18,5,1,35,74,90,56,25,6,1,
%T A348013 56,176,216,179,90,33,7,1,126,352,552,492,315,134,42,8,1,210,794,1269,
%U A348013 1362,966,510,189,52,9,1,462,1588,3033,3480,2890,1716,777,256,63,10,1,792,3473
%N A348013 Triangle by rows: T(n,k) is the number of n-step Dyck  paths with k catastrophes.
%C A348013 T(n,k) is the number chains of k "incomplete" Dyck paths  with a total length of n. (Incomplete Dyck paths are those not ending at the horizontal axis.) Each of the k subsections of the paths does not return to the horizontal axis; they are commonly referred to as paths with catastrophes (like black Fridays on stock market charts).
%F A348013 T(n,1) = A037952(n).
%F A348013 T(n,2) = A191389(n+2).
%F A348013 The generating function of column k is g037952(x)^k, where g037952(x) = x +x^2 +3*x^3+... is the generating function of A037952.
%e A348013 The triangle starts
%e A348013     1
%e A348013     1    1
%e A348013     3    2    1
%e A348013     4    7    3    1
%e A348013    10   14   12    4    1
%e A348013    15   37   31   18    5    1
%e A348013    35   74   90   56   25    6    1
%e A348013    56  176  216  179   90   33    7    1
%e A348013   126  352  552  492  315  134   42    8    1
%e A348013   210  794 1269 1362  966  510  189   52    9    1
%e A348013   462 1588 3033 3480 2890 1716  777  256   63   10    1
%e A348013   792 3473 6781 8901 8060 5521 2835 1130  336   75   11    1
%e A348013 T(1,1)=1 counts U| where the vertical bar indicates starting a new path at the horizontal axis (the catastrophe).
%e A348013 T(2,1)=1 counts UU|.
%e A348013 T(4,1)=4 counts UUUU|, UUUD|, UUDU|, UDUU|.
%e A348013 T(3,2)=2 counts UU|U| and U|UU| .
%e A348013 T(4,2)=7 counts U|UUU|, U|UUD|, U|UDU|, UU|UU|, UUU|U|, UUD|U| and UDU|U|.
%Y A348013 Cf. A348012 (row sums), A037952 (k=1), A191389 (k=2).
%K A348013 nonn,tabl,easy
%O A348013 1,4
%A A348013 _R. J. Mathar_, Sep 24 2021