A348013 - OEIS

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

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OFFSET

1,4

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..68.

FORMULA

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.

EXAMPLE

The triangle starts

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|.

CROSSREFS

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

Sequence in context: A306801 A117212 A208153 * A105033 A092486 A159966

Adjacent sequences: A348010 A348011 A348012 * A348014 A348015 A348016

KEYWORD

nonn,tabl,easy

AUTHOR

R. J. Mathar, Sep 24 2021

STATUS

approved