A348012 -id:A348012 - cp's OEIS frontend

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

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, 56, 176, 216, 179, 90, 33, 7, 1, 126, 352, 552, 492, 315, 134, 42, 8, 1, 210, 794, 1269, 1362, 966, 510, 189, 52, 9, 1, 462, 1588, 3033, 3480, 2890, 1716, 777, 256, 63, 10, 1, 792, 3473

Offset: 1

R. J. Mathar, Sep 24 2021

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).

			The triangle starts
    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
   56  176  216  179   90   33    7    1
  126  352  552  492  315  134   42    8    1
  210  794 1269 1362  966  510  189   52    9    1
  462 1588 3033 3480 2890 1716  777  256   63   10    1
  792 3473 6781 8901 8060 5521 2835 1130  336   75   11    1
T(1,1)=1 counts U| where the vertical bar indicates starting a new path at the horizontal axis (the catastrophe).
T(2,1)=1 counts UU|.
T(4,1)=4 counts UUUU|, UUUD|, UUDU|, UDUU|.
T(3,2)=2 counts UU|U| and U|UU| .
T(4,2)=7 counts U|UUU|, U|UUD|, U|UDU|, UU|UU|, UUU|U|, UUD|U| and UDU|U|.

Cf. A348012 (row sums), A037952 (k=1), A191389 (k=2).

T(n,1) = A037952(n).
T(n,2) = A191389(n+2).
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.

cp's OEIS Frontend

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

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