A171651 -id:A171651 - cp's OEIS frontend

A097692 Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UDUs.

1, 2, 4, 2, 10, 8, 2, 26, 30, 12, 2, 70, 104, 60, 16, 2, 192, 350, 260, 100, 20, 2, 534, 1152, 1050, 520, 150, 24, 2, 1500, 3738, 4032, 2450, 910, 210, 28, 2, 4246, 12000, 14952, 10752, 4900, 1456, 280, 32, 2, 12092, 38214, 54000, 44856, 24192, 8820, 2184, 360, 36, 2

Offset: 0

David Callan, Aug 19 2004; corrected Jun 10 2005

See A091869 for the distribution of the parameter "number of UDUs" on Dyck paths.

			Table begins
\ k 0, 1, 2, ...
n
0 | 1
1 | 2
2 | 4, 2
3 | 10, 8, 2
4 | 26, 30, 12, 2
5 | 70, 104, 60, 16, 2
6 |192, 350, 260, 100, 20, 2
7 |534, 1152, 1050, 520, 150, 24, 2
The path UDUDUD contains 2 UDUs and a(2,1) = 2 because each of UDUD, DUDU contains one UDU.

Aristidis Sapounakis, Panagiotis Tsikouras, Ioannis Tasoulas, Kostas Manes, Strings of Length 3 in Grand-Dyck Paths and the Chung-Feller Property, Electr. J. Combinatorics, 19 (2012), #P2. - From N. J. A. Sloane, Feb 06 2013

Alois P. Heinz, Rows n = 0..141, flattened

Column k=0 is A025565. The row sums are the (even) central binomial coefficients A000984.

Cf. A171651.

b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1,
      expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1))
      +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Apr 29 2015

gfForBalancedByNumberUDU=Sqrt[(1 + x - x*y)/(1 - 3*x - x*y)]; Map[CoefficientList[ #, y]&, CoefficientList[Normal[Series[gfForBalancedByNumberUDU, {x, 0, 8}, {y, 0, 8}]], x]]

G.f.: ((1 + x - x*y)/(1 - 3*x - x*y))^(1/2) = Sum_{n>=0, k>=0} a(n,k) x^n y^k.

Keyword tabl changed to tabf by Michel Marcus, Apr 07 2013

A171670 Triangle T read by rows : T(n,k)= A007318(n,k)*A005773(n-k).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 6, 3, 1, 13, 20, 12, 4, 1, 35, 65, 50, 20, 5, 1, 96, 210, 195, 100, 30, 6, 1, 267, 672, 735, 455, 175, 42, 7, 1, 750, 2136, 2688, 1960, 910, 280, 56, 8, 1, 2123, 6750, 9612, 8064, 4410, 1638, 420, 72, 9, 1

Offset: 0

Philippe Deléham, Dec 14 2009

			Triangle begins : 1 ; 1,1 ; 2,2,1 ; 5,6,3,1 ; 13,20,12,4,1 ; 35,65,50,20,5,1 ; ...

Cf. A171651

Sum_{k, 0<=k<=n} T(n,k)= A024718(n).

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 30, 9, 1, 126, 140, 60, 12, 1, 462, 630, 350, 100, 15, 1, 1716, 2772, 1890, 700, 150, 18, 1, 6435, 12012, 9702, 4410, 1225, 210, 21, 1, 24310, 51480, 48048, 25872, 8820, 1960, 280, 24, 1

Offset: 0

Philippe Deléham, Dec 19 2009

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    35,   30,    9,   1;
   126,  140,   60,  12,   1;
   462,  630,  350, 100,  15,  1;
  1716, 2772, 1890, 700, 150, 18, 1;
  ...

Cf. A107230, A171651

Cf. A001405, A001700, A005573, A005773, A026378, A099323, A122898, A168491.

T[n_,k_]:=n!SeriesCoefficient[Exp[2*x]*(BesselI[0,2*x]+BesselI[1,2*x])*x^k / k!,{x,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Dec 23 2023 *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively.
Conjectural g.f.: 1/(2*t)*( sqrt( (1 - x*t)/(1 - (4 + x)*t) ) - 1 ) = 1 + (3 + x)*t + (10 + 6*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013
E.g.f. of column k: exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 23 2023

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A097692 Triangle read by rows: a(n,k) = number of paths of n upsteps U and n downsteps D that contain k UDUs.

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A171670 Triangle T read by rows : T(n,k)= A007318(n,k)*A005773(n-k).

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A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

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