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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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

A171486 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A033321.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 21, 16, 9, 4, 1, 79, 58, 31, 14, 5, 1, 311, 224, 117, 52, 20, 6, 1, 1265, 900, 465, 205, 80, 27, 7, 1, 5275, 3720, 1910, 840, 330, 116, 35, 8, 1, 22431, 15713, 8034, 3532, 1396, 501, 161, 44, 9, 1, 96900, 67522, 34419, 15136, 6015, 2190
Offset: 0

Views

Author

Philippe Deléham, Dec 09 2009

Keywords

Comments

Equal to B*A065600 = A171224*B where B = A007318 ; equal to B*A039598*B^(-2).

Examples

			Triangle begins :
1
1, 1
2, 2, 1
6, 5, 3, 1
21, 16, 9, 4, 1
79, 58, 31, 14, 5, 1
311, 224, 117, 52, 20, 6, 1

Crossrefs

Cf. A171224, A104259, A091965

Formula

Sum_{k, 0<=k<=n} T(n,k)*x^k = A117641(n), A033321(n), A007317(n+1), A002212(n+1), A026378(n+1) for x = -1, 0, 1, 2, 3 respectively.
T(n,k) = T(n-1,k-1) + T(n-1,k) + sum_{i, i>=0} T(n-1,k+1+i)*2^i. - Philippe Deléham, Feb 23 2012

A191525 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n and having k hills; a hill is a (1,1)-step starting at level 0 and followed by a (1,-1)-step.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 1, 7, 2, 1, 13, 5, 1, 1, 24, 8, 2, 1, 46, 16, 6, 1, 1, 86, 28, 9, 2, 1, 166, 58, 19, 7, 1, 1, 314, 103, 32, 10, 2, 1, 610, 211, 71, 22, 8, 1, 1, 1163, 382, 121, 36, 11, 2, 1, 2269, 781, 261, 85, 25, 9, 1, 1, 4352, 1432, 456, 140, 40, 12, 2, 1, 8518, 2920, 976, 316, 100, 28, 10, 1, 1
Offset: 0

Views

Author

Emeric Deutsch, Jun 06 2011

Keywords

Comments

Row n contains 1+floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
T(n,0)=A191526(n).
Sum(k*T(n,k), k>=0) = A037952(n-1).

Examples

			T(5,1)=2 because we have (UD)UUD and (UD)UUU, where U=(1,1) and D=(1,-1) (the hills are shown between parentheses).
Triangle starts:
  1;
  1;
  1,1;
  2,1;
  4,1,1;
  7,2,1;
  13,5,1,1;

Crossrefs

Cf. A001405, A191526, A037952, A065600.

Programs

  • Maple
    c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := 1/((1-z*c)*(1-z^4*c^2-t*z^2)): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Formula

G.f.: G(t,z)=1/((1-z*c)*(1-z^4*c^2-t*z^2)), where c=(1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2.
Previous Showing 11-13 of 13 results.

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

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