cp's OEIS Frontend

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.

%I A064044 #21 Feb 26 2015 08:01:06
%S A064044 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 A064044 200,234,108,30,6,1,0,35,700,1110,624,195,42,7,1,0,70,2450,5460,3760,
%U A064044 1350,318,56,8,1,0,126,8820,27405,23480,9770,2556,483,72,9,1,0,252
%N 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.
%C A064044 E.g.f. of row n equals ( besseli(0,2*y) + y*besseli(1,2*y) )^n. - _Paul D. Hanna_, Apr 07 2005
%H A064044 Alois P. Heinz, <a href="/A064044/b064044.txt">Antidiagonals n = 0..140, flattened</a>
%H A064044 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 A064044 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 A064044 E.g.f: 1/(1 - x*besseli(0, 2*y) - x*y*besseli(1, 2*y)). - _Paul D. Hanna_, Apr 07 2005
%e A064044 Rows start:
%e A064044 1, 0,  0,   0,    0,     0,      0, ...
%e A064044 1, 1,  2,   3,    6,    10,     20, ...
%e A064044 1, 2,  6,  18,   60,   200,    700, ...
%e A064044 1, 3, 12,  51,  234,  1110,   5460, ...
%e A064044 1, 4, 20, 108,  624,  3760,  23480, ...
%e A064044 1, 5, 30, 195, 1350,  9770,  73300, ...
%e A064044 1, 6, 42, 318, 2556, 21480, 187140, ...
%p A064044 a:= proc(n, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
%p A064044        add(binomial(k, j)*binomial(j, floor(j/2))
%p A064044        *a(n-1, k-j), j=0..k))
%p A064044     end:
%p A064044 seq(seq(a(n,d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, May 06 2014
%t A064044 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 A064044 (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 A064044 Rows include A000007, A001405, A005566, A064036. Columns include A000012, A001477, A002378, A064043. Cf. A064045.
%K A064044 nonn,tabl
%O A064044 0,8
%A A064044 _Henry Bottomley_, Aug 23 2001