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A295987
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Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step patterns 010 or 101, where 1=up and 0=down; triangle T(n,k), n >= 0, k = max(0, n-3), read by rows.
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11
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1, 1, 2, 6, 14, 10, 52, 36, 32, 204, 254, 140, 122, 1010, 1368, 1498, 620, 544, 5466, 9704, 9858, 9358, 3164, 2770, 34090, 67908, 90988, 72120, 63786, 18116, 15872, 233026, 545962, 762816, 839678, 560658, 470262, 115356, 101042, 1765836, 4604360, 7458522
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OFFSET
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0,3
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LINKS
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EXAMPLE
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Triangle T(n,k) begins:
: 1;
: 1;
: 2;
: 6;
: 14, 10;
: 52, 36, 32;
: 204, 254, 140, 122;
: 1010, 1368, 1498, 620, 544;
: 5466, 9704, 9858, 9358, 3164, 2770;
: 34090, 67908, 90988, 72120, 63786, 18116, 15872;
: 233026, 545962, 762816, 839678, 560658, 470262, 115356, 101042;
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MAPLE
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b:= proc(u, o, t, h) option remember; expand(
`if`(u+o=0, 1, `if`(t=0, add(b(u-j, j-1, 1$2), j=1..u),
add(`if`(h=3, x, 1)*b(u-j, o+j-1, [1, 3, 1][t], 2), j=1..u)+
add(`if`(t=3, x, 1)*b(u+j-1, o-j, 2, [1, 3, 1][h]), j=1..o))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=0..12);
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MATHEMATICA
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b[u_, o_, t_, h_] := b[u, o, t, h] = Expand[If[u + o == 0, 1, If[t == 0, Sum[b[u - j, j - 1, 1, 1], {j, 1, u}], Sum[If[h == 3, x, 1]*b[u - j, o + j - 1, {1, 3, 1}[[t]], 2], {j, 1, u}] + Sum[If[t == 3, x, 1]*b[u + j - 1, o - j, 2, {1, 3, 1}[[h]]], {j, 1, o}]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 0, 0]];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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