A191389 -id:A191389 - cp's OEIS frontend

A191387 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k valleys at level 0.

1, 1, 2, 3, 5, 1, 8, 2, 14, 5, 1, 23, 10, 2, 41, 22, 6, 1, 69, 42, 13, 2, 125, 87, 32, 7, 1, 214, 164, 66, 16, 2, 393, 330, 149, 43, 8, 1, 682, 618, 301, 94, 19, 2, 1267, 1225, 648, 227, 55, 9, 1, 2223, 2288, 1290, 484, 126, 22, 2, 4171, 4498, 2700, 1100, 322, 68, 10, 1, 7385, 8396, 5322, 2300, 718, 162, 25, 2

Offset: 0

Emeric Deutsch, Jun 02 2011

A dispersed Dyck paths of length n is a Motzkin path of length n with no (1,0) steps at positive heights.
Row n >=2 has floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
T(n,0) = A191388(n).
Sum_{k>=0} k*T(n,k) = A191389(n).

			T(5,1)=2 because we have HUDUD and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
   1;
   1;
   2;
   3;
   5,  1;
   8,  2;
  14,  5,  1;
  23, 10,  2;
  41, 22,  6,  1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A001405, A191388, A191389.

G := (1+z^2*c-t*z^2*c)/(1-z-z^3*c-t*z^2*c*(1-z)): c := ((1-sqrt(1-4*z^2))*1/2)/z^2: Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)-1) end do; # yields sequence in triangular form

G.f.: G=G(t,z) is given by G = 1 + z*G + z^2*c*(t*(G-1-z*G) + 1 + z*G), where c = (1-sqrt(1-4*z^2))/(2*z^2).

A123514 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 321 exactly once (n>=3; 1<=k<=n-2).

Original entry on oeis.org

1, 0, 2, 4, 0, 3, 0, 10, 0, 4, 14, 0, 18, 0, 5, 0, 40, 0, 28, 0, 6, 48, 0, 81, 0, 40, 0, 7, 0, 150, 0, 140, 0, 54, 0, 8, 165, 0, 330, 0, 220, 0, 70, 0, 9, 0, 550, 0, 616, 0, 324, 0, 88, 0, 10, 572, 0, 1287, 0, 1040, 0, 455, 0, 108, 0, 11, 0, 2002, 0, 2548, 0, 1638, 0, 616, 0, 130, 0, 12

Offset: 3

Emeric Deutsch, Oct 13 2006

			T(4,2)=2 because we have 1432 and 3214 (also 4231 is an involution with 2 fixed points but contains twice the pattern 321: 421 and 431).
Triangle starts:
    1;
    0,   2;
    4,   0,    3;
    0,  10,    0,   4;
   14,   0,   18,   0,    5;
    0,  40,    0,  28,    0,   6;
   48,   0,   81,   0,   40,   0,   7;
    0, 150,    0, 140,    0,  54,   0,  8;
  165,   0,  330,   0,  220,   0,  70,  0,   9;
    0, 550,    0, 616,    0, 324,   0, 88,   0, 10;
  572,   0, 1287,   0, 1040,   0, 455,  0, 108,  0, 11;

G. C. Greubel, Rows n = 3..53 of the triangle, flattened
E. Deutsch, A. Robertson and D. Saracino, Refined restricted involutions, European Journal of Combinatorics 28 (2007), 481-498 (see pp. 493 and 498).

Cf. A003517, A112554, A123515, A191389.

A123514:= func< n,k | ((1+(-1)^(n-k))/(2*(n+1)))*k*(k+3)*Binomial(n+1, Floor((n-k-2)/2)) >;
[A123514(n,k): k in [1..n-2], n in [3..15]]; // G. C. Greubel, Jan 15 2022

T:=proc(n,k) if n-k mod 2 = 0 and k<=n then k*(k+3)*binomial(n+1,(n-k)/2-1)/(n+1) else 0 fi end: for n from 3 to 15 do seq(T(n,k),k=1..n-2) od; # yields sequence in triangular form

T[n_, k_]:= ((1+(-1)^(n-k))/2)*k*(k+3)*Binomial[n+1, (n-k-2)/2]/(n+1);
Table[T[n, k], {n, 3, 15}, {k, n-2}]//Flatten (* G. C. Greubel, Jan 15 2022 *)

def A123514(n,k): return ((1+(-1)^(n-k))/(2*(n+1)))*k*(k+3)*binomial(n+1, (n-k-2)//2)
flatten([[A123514(n,k) for k in (1..n-2)] for n in (3..15)]) # G. C. Greubel, Jan 15 2022

T(n,k) = k*(k+3)*binomial(n+1,(n-k-2)/2)/(n+1), for n>=3, 1<=k<=n-2, n-k even.
From G. C. Greubel, Jan 15 2022: (Start)
Sum_{k=1..n-2} T(n, k) = A191389(n+1).
Sum_{k=1..floor((n-1)/2)} T(n-k, k) = ((1-(-1)^n)/2)*(12/(n+9))*binomial(n+2, (n- 3)/2). (End)

A191397 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having k DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 1, 18, 2, 28, 7, 56, 14, 89, 37, 179, 72, 1, 289, 170, 3, 585, 326, 13, 956, 726, 34, 1948, 1380, 104, 3214, 2970, 250, 1, 6591, 5616, 659, 4, 10959, 11829, 1502, 20, 22609, 22300, 3647, 64, 37833, 46306, 8019, 220, 78486, 87154, 18495, 620, 1, 132037, 179222, 39648, 1804, 5

Offset: 0

Emeric Deutsch, Jun 04 2011

Row n has 1+floor(n/5) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
T(n,0) = A191398(n).
Sum_{k>=0} k*T(n,k) = A191389(n-1).

			T(6,1)=2 because we have HU(DHU)D and U(DHU)DH, where U=(1,1), D=(1,-1), H=(1,0) (the DHU's are shown between parentheses).
Triangle starts:
   1;
   1;
   2;
   3;
   6;
   9,  1;
  18,  2;
  28,  7;
  56, 14;

Cf. A001405, A191389, A191398.

G := 2/(1-z-2*z^3-t*z+2*t*z^3+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 21 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 21 do seq(coeff(P[n], t, k), k = 0 .. floor((1/5)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) = 2/(1-z-2*z^3-t*z+2*t*z^3+(1-z+t*z)*sqrt(1-4*z^2)).

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

Original entry on oeis.org

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.

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A191387 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k valleys at level 0.

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A123514 Triangle read by rows: T(n,k) is the number of involutions of {1,2,...,n} with exactly k fixed points and which contain the pattern 321 exactly once (n>=3; 1<=k<=n-2).

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A191397 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having k DHU's (here U=(1,1), H=(1,0), and D=(1,-1)).

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

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