cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A243752 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1); triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 3, 1, 1, 11, 2, 9, 16, 12, 4, 1, 1, 57, 69, 5, 127, 161, 98, 35, 7, 1, 323, 927, 180, 1515, 1997, 1056, 280, 14, 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1, 10455, 25638, 18357, 4115, 220, 1, 20705, 68850, 77685, 34840, 5685, 246, 1
Offset: 0

Views

Author

Alois P. Heinz, Jun 09 2014

Keywords

Examples

			Triangle T(n,k) begins:
: n\k :    0     1     2     3    4    5  ...
+-----+----------------------------------------------------------
:  0  :    1;                                 [row  0 of A131427]
:  1  :    0,    1;                           [row  1 of A131427]
:  2  :    0,    1,    1;                     [row  2 of A090181]
:  3  :    1,    3,    1;                     [row  3 of A001263]
:  4  :    1,   11,    2;                     [row  4 of A091156]
:  5  :    9,   16,   12,    4,   1;          [row  5 of A091869]
:  6  :    1,   57,   69,    5;               [row  6 of A091156]
:  7  :  127,  161,   98,   35,   7,   1;     [row  7 of A092107]
:  8  :  323,  927,  180;                     [row  8 of A091958]
:  9  : 1515, 1997, 1056,  280,  14;          [row  9 of A135306]
: 10  : 4191, 5539, 3967, 1991, 781, 244, ... [row 10 of A094507]

Links

Crossrefs

Columns k=0-10 give: A243754, A243770, A243771, A243772, A243773, A243774, A243775, A243776, A243777, A243778, A243779, or main diagonals of A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.
Row sums give A000108.
Cf. A098978, A114463, A114848, A116424, A135305, A242450, A243366, A243838.

A171380 Expansion of the first column of triangle T_(1,x), T(x,y) defined in A039599; T_(1,0)= A061554, T_(1,1)= A064189, T_(1,2)= A039599, T_(1,3)= A110877, T_(1,4)= A124576.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 6, 2, 1, 0, 0, 10, 8, 2, 1, 0, 0, 20, 16, 12, 2, 1, 0, 0, 35, 47, 25, 17, 2, 1, 0, 0, 70, 94, 97, 36, 23, 2, 1, 0, 0, 126, 244, 204, 179, 49, 30, 2, 1, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 07 2009

Keywords

Comments

Diagonal sums: A089324.
Equal to A092107*B^(-1) = A092107*A130595 as lower triangular arrays. - Philippe Deléham, Dec 10 2009

Examples

			Triangle begins:
   1;
   1, 0;
   2, 0, 0;
   3, 1, 0, 0;
   6, 2, 1, 0, 0;
  10, 8, 2, 1, 0, 0;
  ...

Crossrefs

Cf. A000108, A001006, A001405, A033321, A089324.

Formula

Sum_{k=0..n} T(n,k)*x^k = A001405(n), A001006(n), A000108(n), A033321(n) for x = 0, 1, 2, 3 respectively.

A098747 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having exactly k UDU's at low level.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 24, 11, 5, 1, 1, 75, 35, 14, 6, 1, 1, 243, 113, 47, 17, 7, 1, 1, 808, 376, 156, 60, 20, 8, 1, 1, 2742, 1276, 532, 204, 74, 23, 9, 1, 1, 9458, 4402, 1840, 712, 257, 89, 26, 10, 1, 1, 33062, 15390, 6448, 2507, 917, 315, 105, 29, 11, 1, 1, 116868
Offset: 1

Views

Author

N. J. A. Sloane, Oct 30 2004

Keywords

Comments

T(n,0) = A000958(n-1). - Emeric Deutsch, Dec 23 2006

Examples

			Triangle begins:
1
1 1
3 1 1
8 4 1 1
24 11 5 1 1
75 35 14 6 1 1
T(4,2)=1 because we have UDUDUUDD.

Links

Crossrefs

Cf. A091869, A092107.
Cf. A000958.

Programs

  • Maple
    c:=(1-sqrt(1-4*z))/2/z: G:=z*c/(1-t*z+z-z*c): Gser:=simplify(series(G,z=0,15)): for n from 1 to 13 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 12 do seq(coeff(P[n],t,k),k=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch, Dec 23 2006
  • Mathematica
    u[n_, k_, i_]:=(2i+1)/(n-k)Binomial[k+i, i]Binomial[2n-2k-2i-2, n-k-1] u[n_, k_]/;k<=n-1 := Sum[u[n, k, i], {i, 0, n-k-1}] Table[u[n, k], {n, 10}, {k, 0, n-1}] (* u[n, k, i] is the number of Dyck n-paths with k low UDUs and k+i+1 returns altogether. For example, with n=4, k=1 and i=1, u[n, k, i] counts UDUUDDUD, UUDDUDUD because each has size n=4, k=1 low UDUs and k+i+1=3 returns to ground level. *) (* David Callan, Nov 03 2005 *)

Formula

See Mathematica line.
G.f.=zC/(1+z-tz-zC), where C=(1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch, Dec 23 2006
With offset 0 (0<=k<=n), T(n,k)=A065600(n,k)+A065600(n+1,k)-A065600(n,k-1). - Philippe Deléham, Apr 01 2007
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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