A276418 Starting a random walk on Z at 0 triangle T(j,k) gives the number of paths of length 2*j returning to 0 exactly k times.

1, 2, 2, 6, 6, 4, 20, 20, 16, 8, 70, 70, 60, 40, 16, 252, 252, 224, 168, 96, 32, 924, 924, 840, 672, 448, 224, 64, 3432, 3432, 3168, 2640, 1920, 1152, 512, 128, 12870, 12870, 12012, 10296, 7920, 5280, 2880, 1152, 256, 48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512, 184756, 184756, 175032, 155584, 128128, 96096, 64064, 36608, 16896, 5632, 1024

Offset: 0

Author

Franz Vrabec, Sep 27 2016

Keywords

• nonn
• tabl
• walk
• easy

Comments

The number of paths of odd length 2*j+1 is the same as the number of even length 2*j (returning to 0 exactly k times).

Examples

			Triangle T(j,k) begins:
      1
      2,     2
      6,     6,     4
     20,    20,    16,     8
     70,    70,    60,    40,    16
    252,   252,   224,   168,    96,    32
    924,   924,   840,   672,   448,   224,    64
   3432,  3432,  3168,  2640,  1920,  1152,   512,  128
  12870, 12870, 12012, 10296,  7920,  5280,  2880, 1152,  256
  48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..1325

Programs

GAP

Flat(List([0..10],j->List([0..j],k->2^k*Binomial(2*j-k,j-k)))); # Muniru A Asiru, May 18 2018

Formula

T(j,k) = (2^k)*C(2*j-k,j-k).

T(j,0) = T(j,1) for j>0.

T(j,0) = A000984(j).

T(j,1) = A000984(j) for j>0.

T(j,2) = A128650(j+1).

T(j,j) = A000079(j).

T(j,j-1) = A057711(j+1) for j>0.