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A064045 Square array read by antidiagonals of number of length 2k walks on an n-dimensional hypercubic lattice starting and finishing at the origin and staying in the nonnegative part. 3
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 10, 3, 1, 0, 14, 70, 24, 4, 1, 0, 42, 588, 285, 44, 5, 1, 0, 132, 5544, 4242, 740, 70, 6, 1, 0, 429, 56628, 73206, 16016, 1525, 102, 7, 1, 0, 1430, 613470, 1403028, 410928, 43470, 2730, 140, 8, 1, 0, 4862, 6952660, 29082339, 11925672, 1491210, 96684, 4445, 184, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

FORMULA

a(n,k) = Sum_{j=0..k} C(2k,2j) c(j) a(n-1,k-j) where c(j) = C(2j,j)/(j+1) = A000108(j) with a(0,0) = 1 and a(0,k) = 0 for k>0.

EXAMPLE

Rows start:

1, 0,  0,   0,    0,     0,       0, ...

1, 1,  2,   5,   14,    42,     132, ...

1, 2, 10,  70,  588,  5544,   56628, ...

1, 3, 24, 285, 4242, 73206, 1403028, ...

MAPLE

a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),

       add(binomial(2*k, 2*j)*binomial(2*j, j)/

       (j+1)*a(n-1, k-j), j=0..k))

    end:

seq(seq(a(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, May 06 2014

MATHEMATICA

a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[2*k, 2*j]* Binomial[2*j, j]/(j+1)*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Feb 26 2015, after Alois P. Heinz *)

CROSSREFS

Rows include A000007, A000108, A005568, A064037. Columns include A000012, A001477, A049450, A064046. Cf. A064044.

Sequence in context: A242153 A065066 A266291 * A258829 A110314 A152882

Adjacent sequences:  A064042 A064043 A064044 * A064046 A064047 A064048

KEYWORD

nonn,tabl

AUTHOR

Henry Bottomley, Aug 23 2001

STATUS

approved

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Last modified December 10 15:13 EST 2016. Contains 279003 sequences.