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.
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
Offset: 0
Examples
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, ...
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
Crossrefs
Programs
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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: 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[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 *) -
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 */
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
Comments