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A334622
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A(n,k) is the sum of the k-th powers of the descent set statistics for permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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7
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1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 6, 8, 1, 1, 2, 10, 24, 16, 1, 1, 2, 18, 88, 120, 32, 1, 1, 2, 34, 360, 1216, 720, 64, 1, 1, 2, 66, 1576, 14460, 24176, 5040, 128, 1, 1, 2, 130, 7224, 190216, 994680, 654424, 40320, 256, 1, 1, 2, 258, 34168, 2675100, 46479536, 109021500, 23136128, 362880, 512
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OFFSET
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0,6
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LINKS
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FORMULA
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A(n,k) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^k.
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
2, 2, 2, 2, 2, 2, 2, ...
4, 6, 10, 18, 34, 66, 130, ...
8, 24, 88, 360, 1576, 7224, 34168, ...
16, 120, 1216, 14460, 190216, 2675100, 39333016, ...
32, 720, 24176, 994680, 46479536, 2368873800, 128235838496, ...
...
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MAPLE
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b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
add(b(u-j, o+j-1, t+1)*x^floor(2^(t-1)), j=1..u)+
add(b(u+j-1, o-j, t+1), j=1..o)))
end:
A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1,
Sum[b[u - j, o + j - 1, t + 1] x^Floor[2^(t - 1)], {j, 1, u}] +
Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]]];
A[n_, k_] := Function[p, Sum[Coefficient[p, x, i]^k, {i, 0, Exponent[p, x]}]][b[n, 0, 0]];
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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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