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A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A361397 #46 Mar 21 2023 13:26:19
%S A361397 1,1,0,1,2,0,1,4,2,0,1,6,20,4,0,1,8,54,176,10,0,1,10,104,996,1876,28,
%T A361397 0,1,12,170,2944,22734,22064,84,0,1,14,252,6500,108136,577692,275568,
%U A361397 264,0,1,16,350,12144,332050,4525888,15680628,3584064,858,0
%N A361397 Number A(n,k) of k-dimensional cubic lattice walks with 2n steps from origin to origin and avoiding early returns to the origin; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A361397 Column k is INVERTi transform of k-th row of A287318.
%H A361397 Alois P. Heinz, <a href="/A361397/b361397.txt">Antidiagonals n = 0..140, flattened</a>
%F A361397 A(n,1)/2 = A000108(n-1) for n >= 1.
%F A361397 G.f. of column k: 2 - 1/Integral_{t=0..oo} exp(-t)*BesselI(0,2*t*sqrt(x))^k dt. - _Shel Kaphan_, Mar 19 2023
%e A361397 Square array A(n,k) begins:
%e A361397   1,  1,     1,      1,       1,        1,        1, ...
%e A361397   0,  2,     4,      6,       8,       10,       12, ...
%e A361397   0,  2,    20,     54,     104,      170,      252, ...
%e A361397   0,  4,   176,    996,    2944,     6500,    12144, ...
%e A361397   0, 10,  1876,  22734,  108136,   332050,   796860, ...
%e A361397   0, 28, 22064, 577692, 4525888, 19784060, 62039088, ...
%p A361397 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A361397       add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
%p A361397     end:
%p A361397 g:= proc(n, k) option remember; `if` (n<1, -1,
%p A361397       -add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n))
%p A361397     end:
%p A361397 A:= (n,k)-> `if`(n=0, 1, `if`(k=0, 0, g(n, k))):
%p A361397 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A361397 b[n_, 0] = 0; b[n_, 1] = 1; b[0, k_] = 1;
%t A361397 b[n_, k_] := b[n, k] = Sum[Binomial[n, i]^2*b[i, k - 1], {i, 0, n}]; (* A287316 *)
%t A361397 g[n_, k_] := g[n, k] = b[n, k]*Binomial[2 n, n]; (* A287318 *)
%t A361397 a[n_, k_] := a[n, k] = g[n, k] - Sum[a[i, k]*g[n - i, k], {i, 1, n - 1}];
%t A361397 TableForm[Table[a[n, k], {k, 0, 10}, {n, 0, 10}]] (* _Shel Kaphan_, Mar 14 2023 *)
%Y A361397 Columns k=0-5 give: A000007, |A002420|, A054474, A049037, A359801, A361364.
%Y A361397 Rows n=0-2 give: A000012, A005843, A139271.
%Y A361397 Main diagonal gives A361297.
%Y A361397 Cf. A000108, A287318.
%K A361397 nonn,tabl
%O A361397 0,5
%A A361397 _Alois P. Heinz_, Mar 10 2023