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A227584
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Number of lattice paths from {4}^n to {0}^n using steps that decrement one component such that for each point (p_1,p_2,...,p_n) we have p_1<=p_2<=...<=p_n.
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2
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1, 8, 185, 7680, 456033, 34426812, 3086989927, 315051017342, 35566911169298, 4353511908566248, 569413385415535738, 78713723425497511522, 11403561640157735499129, 1719932910431380877877228, 268627766543783314569921051, 43259068400832620021992394382
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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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Recurrence: 2*(n+2)^3*(n+3)*(3*n + 4)*(3*n + 5)*(21546*n^6 - 45513*n^5 - 3699*n^4 + 13101*n^3 + 6745*n^2 + 2032*n + 228)*a(n) = (n+2)*(100641366*n^11 + 49968333*n^10 - 359008281*n^9 - 154345878*n^8 + 328506480*n^7 - 50516019*n^6 - 255412581*n^5 + 41482508*n^4 + 116397064*n^3 + 50875696*n^2 + 13946672*n + 1423680)*a(n-1) - 2*(173359116*n^12 + 510099768*n^11 - 127771911*n^10 - 1779706188*n^9 - 1320942528*n^8 + 1504634418*n^7 + 2094287811*n^6 + 74489810*n^5 - 783633768*n^4 - 274711936*n^3 + 13818000*n^2 + 12588128*n + 862080)*a(n-2) - 16*(2*n - 3)*(2*n + 1)*(3*n - 2)*(3*n - 1)*(4*n + 1)*(4*n + 3)*(21546*n^6 + 83763*n^5 + 91926*n^4 - 25905*n^3 - 108086*n^2 - 58260*n - 5560)*a(n-3).
a(n) ~ 2^(8*n+51/2) / (3^11 * Pi^(3/2) * n^(15/2)).
(End)
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EXAMPLE
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a(1) = 8: [(4),(0)], [(4),(1),(0)], [(4),(2),(0)], [(4),(2),(1),(0)], [(4),(3),(0)], [(4),(3),(1),(0)], [(4),(3),(2),(0)], [(4),(3),(2),(1),(0)].
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MAPLE
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b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
end:
a:= n-> `if`(n=0, 1, b([4$n])):
seq(a(n), n=0..16);
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MATHEMATICA
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b[l_] := b[l] = If[l[[-1]] == 0, 1, Sum[Sum[b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i - 1]]], l[[i]] - 1}], {i, 1, Length[l]}]];
a[n_] := If[n == 0, 1, b[Array[4&, n]]];
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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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