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A286918
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Total number of nodes summed over all lattice paths from (0,0) to (n,n) using steps {(k,0), (0,k) | 0<k<=4} which never go above the diagonal x=y.
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2
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1, 3, 21, 159, 1257, 10046, 81811, 674184, 5605141, 46920874, 394949193, 3339464105, 28343082002, 241324470723, 2060357315568, 17632454524499, 151211115930880, 1299121121946684, 11179500360964780, 96344672766997340, 831385476662968094, 7182777882757416692
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ c * d^n / sqrt(n), where d = 8.84734830841870961487278801886633962039798... is the real root of the equation 4 + 4*d - 8*d^2 - 8*d^3 + d^4 = 0 and c = 0.43633259077568249345422000202799136319250347607927734138960545201547... - Vaclav Kotesovec, May 30 2017
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MAPLE
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b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
`if`(x=0, [1$2], add((p-> p+[0, p[1]])(
b(x-j, y)+b(x, y-j)), j=1..4)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..25);
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MATHEMATICA
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b[x_, y_] := b[x, y] = If[y > x || y < 0, {0, 0}, If[x == 0, {1, 1}, Sum[Function[p, p + {0, p[[1]]}][b[x-j, y] + b[x, y-j]], {j, 1, 4}]]];
a[n_] := b[n, n][[2]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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