|
| |
|
|
A064044
|
|
Square array read by antidiagonals of number of length k walks on an n-dimensional hypercubic lattice starting at the origin and staying in the nonnegative part.
|
|
4
|
|
|
|
1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 6, 3, 1, 0, 6, 18, 12, 4, 1, 0, 10, 60, 51, 20, 5, 1, 0, 20, 200, 234, 108, 30, 6, 1, 0, 35, 700, 1110, 624, 195, 42, 7, 1, 0, 70, 2450, 5460, 3760, 1350, 318, 56, 8, 1, 0, 126, 8820, 27405, 23480, 9770, 2556, 483, 72, 9, 1, 0, 252
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,8
|
|
|
COMMENTS
|
E.g.f. of row n equals ( besseli(0,2*y) + y*besseli(1,2*y) )^n. - Paul D. Hanna, Apr 07 2005
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n,k) = Sum{j=0..k} C(k,j) B(j) a(n-1,k-j) where B(j) = C(j,[j/2]) = A001405(j) with a(0,0) = 1 and a(0,k) = 0 for k>0.
E.g.f: 1/(1 - x*besseli(0, 2*y) - x*y*besseli(1, 2*y)). - Paul D. Hanna, Apr 07 2005
|
|
|
EXAMPLE
|
Rows start:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 2, 3, 6, 10, 20, ...
1, 2, 6, 18, 60, 200, 700, ...
1, 3, 12, 51, 234, 1110, 5460, ...
1, 4, 20, 108, 624, 3760, 23480, ...
1, 5, 30, 195, 1350, 9770, 73300, ...
1, 6, 42, 318, 2556, 21480, 187140, ...
|
|
|
MAPLE
|
a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
add(binomial(k, j)*binomial(j, floor(j/2))
*a(n-1, k-j), j=0..k))
end:
|
|
|
MATHEMATICA
|
a[n_, k_] := a[n, k] = If[n == 0, If[k == 0, 1, 0], Sum[Binomial[k, j]*Binomial[j, Floor[j/2]]*a[n-1, k-j], {j, 0, k}]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
|
|
|
PROG
|
(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); k!*polcoeff(polcoeff(1/(1-X*besseli(0, 2*Y)-X*Y*besseli(1, 2*Y)), n, x), k, y)} /* Hanna */
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
