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A294220
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Number A(n,k) of ascent sequences of length n where no letter multiplicity is larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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5
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1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 10, 1, 0, 1, 1, 2, 5, 14, 27, 1, 0, 1, 1, 2, 5, 15, 47, 83, 1, 0, 1, 1, 2, 5, 15, 52, 180, 277, 1, 0, 1, 1, 2, 5, 15, 53, 210, 773, 1015, 1, 0, 1, 1, 2, 5, 15, 53, 216, 964, 3701, 4007, 1, 0
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OFFSET
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0,13
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LINKS
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FORMULA
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A(n,k) = Sum_{j=0..k} A294219(n,j).
A(n,k) = A(n,n) = A022493(n) for k >= n.
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EXAMPLE
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A(4,2) = 10: 0123, 0011, 0012, 0101, 0102, 0110, 0112, 0120, 0121, 0122.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, ...
0, 1, 4, 5, 5, 5, 5, 5, 5, ...
0, 1, 10, 14, 15, 15, 15, 15, 15, ...
0, 1, 27, 47, 52, 53, 53, 53, 53, ...
0, 1, 83, 180, 210, 216, 217, 217, 217, ...
0, 1, 277, 773, 964, 1006, 1013, 1014, 1014, ...
0, 1, 1015, 3701, 4960, 5270, 5326, 5334, 5335, ...
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MAPLE
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b:= proc(n, i, t, p, k) option remember; `if`(n=0, 1,
add(`if`(coeff(p, x, j)=k, 0, b(n-1, j, t+
`if`(j>i, 1, 0), p+x^j, k)), j=1..t+1))
end:
A:= (n, k)-> b(n, 0$3, min(n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n == 0, 1, Sum[ If[ Coefficient[p, x, j] == k, 0, b[n-1, j, t + If[j>i, 1, 0], p + x^j, k]], {j, 1, t+1}]];
A[n_, k_] := b[n, 0, 0, 0, Min[n, k]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Aug 05 2018, translated from Maple *)
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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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