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A361397
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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.
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7
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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, 0, 1, 12, 170, 2944, 22734, 22064, 84, 0, 1, 14, 252, 6500, 108136, 577692, 275568, 264, 0, 1, 16, 350, 12144, 332050, 4525888, 15680628, 3584064, 858, 0
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history;
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OFFSET
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0,5
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COMMENTS
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Column k is INVERTi transform of k-th row of A287318.
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LINKS
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FORMULA
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A(n,1)/2 = A000108(n-1) for n >= 1.
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
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 2, 4, 6, 8, 10, 12, ...
0, 2, 20, 54, 104, 170, 252, ...
0, 4, 176, 996, 2944, 6500, 12144, ...
0, 10, 1876, 22734, 108136, 332050, 796860, ...
0, 28, 22064, 577692, 4525888, 19784060, 62039088, ...
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
end:
g:= proc(n, k) option remember; `if` (n<1, -1,
-add(g(n-i, k)*(2*i)!*b(i, k)/i!^2, i=1..n))
end:
A:= (n, k)-> `if`(n=0, 1, `if`(k=0, 0, g(n, k))):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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b[n_, 0] = 0; b[n_, 1] = 1; b[0, k_] = 1;
b[n_, k_] := b[n, k] = Sum[Binomial[n, i]^2*b[i, k - 1], {i, 0, n}]; (* A287316 *)
g[n_, k_] := g[n, k] = b[n, k]*Binomial[2 n, n]; (* A287318 *)
a[n_, k_] := a[n, k] = g[n, k] - Sum[a[i, k]*g[n - i, k], {i, 1, n - 1}];
TableForm[Table[a[n, k], {k, 0, 10}, {n, 0, 10}]] (* Shel Kaphan, Mar 14 2023 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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