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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.
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%I #21 Feb 26 2015 08:01:06

%S 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,

%T 200,234,108,30,6,1,0,35,700,1110,624,195,42,7,1,0,70,2450,5460,3760,

%U 1350,318,56,8,1,0,126,8820,27405,23480,9770,2556,483,72,9,1,0,252

%N 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.

%C E.g.f. of row n equals ( besseli(0,2*y) + y*besseli(1,2*y) )^n. - _Paul D. Hanna_, Apr 07 2005

%H Alois P. Heinz, <a href="/A064044/b064044.txt">Antidiagonals n = 0..140, flattened</a>

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

%F 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.

%F E.g.f: 1/(1 - x*besseli(0, 2*y) - x*y*besseli(1, 2*y)). - _Paul D. Hanna_, Apr 07 2005

%e Rows start:

%e 1, 0, 0, 0, 0, 0, 0, ...

%e 1, 1, 2, 3, 6, 10, 20, ...

%e 1, 2, 6, 18, 60, 200, 700, ...

%e 1, 3, 12, 51, 234, 1110, 5460, ...

%e 1, 4, 20, 108, 624, 3760, 23480, ...

%e 1, 5, 30, 195, 1350, 9770, 73300, ...

%e 1, 6, 42, 318, 2556, 21480, 187140, ...

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

%p add(binomial(k, j)*binomial(j, floor(j/2))

%p *a(n-1, k-j), j=0..k))

%p end:

%p seq(seq(a(n,d-n), n=0..d), d=0..12); # _Alois P. Heinz_, May 06 2014

%t 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_ *)

%o (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 */

%Y Rows include A000007, A001405, A005566, A064036. Columns include A000012, A001477, A002378, A064043. Cf. A064045.

%K nonn,tabl

%O 0,8

%A _Henry Bottomley_, Aug 23 2001