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A358612
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Irregular table T(n, k), n >= 0, k > 0, read by rows of extended (due to binary expansion of n) Stirling numbers of the second kind.
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2
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1, 1, 1, 3, 1, 1, 5, 2, 1, 7, 6, 1, 1, 9, 4, 1, 11, 11, 2, 1, 13, 15, 3, 1, 15, 25, 10, 1, 1, 17, 8, 1, 19, 21, 4, 1, 21, 28, 6, 1, 23, 44, 19, 2, 1, 25, 39, 9, 1, 27, 58, 27, 3, 1, 29, 68, 34, 4, 1, 31, 90, 65, 15, 1, 1, 33, 16, 1, 35, 41, 8, 1, 37, 54, 12, 1
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OFFSET
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1,4
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COMMENTS
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Experiments with WolframAlpha lead us to conjecture (which we subsequently check on a large number of values) that
U(n,m) = Sum_{k=1..wt(n) + 2} k!*k^(m+1)*R(n, k)*(-1)^(wt(n) - k + 2) for n > 0, m >= 0 where wt(n) = A000120(n) and where R(n, k) are unknown coefficients.
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LINKS
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FORMULA
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T(n, 1) = 1 for n > 0 with T(0, 1) = T(0, 2) = 1.
T(2n+1, k) = k*T(n, k) + T(n, k-1) for n >= 0, k > 1.
T(2n, k) = k*T(n, k) + T(n, k-1) - (T(2n, k-1) + T(n, k-1))/(k-1) for n > 0, k > 1.
T(2^n - 1, k) = Stirling2(n+2, k) for n >= 0, k > 0.
T(n, 2) = 2n+1 for n >= 0.
Sum_{k=1..wt(n) + 2} k!*T(n, k)*(-1)^(wt(n) - k + 2) = A329369(n) for n >= 0 where wt(n) = A000120(n).
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EXAMPLE
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Irregular table begins:
1, 1;
1, 3, 1;
1, 5, 2;
1, 7, 6, 1;
1, 9, 4;
1, 11, 11, 2;
1, 13, 15, 3;
1, 15, 25, 10, 1;
1, 17, 8;
1, 19, 21, 4;
1, 21, 28, 6;
1, 23, 44, 19, 2;
1, 25, 39, 9;
1, 27, 58, 27, 3;
1, 29, 68, 34, 4;
1, 31, 90, 65, 15, 1;
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PROG
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(PARI) T(n, k)=if(n==0 || k==1, (n==0 && k<3) + (k==1 && n>0), k*T(n\2, k) + T(n\2, k-1) - if(n%2==0, (T(n, k-1) + T(n\2, k-1))/(k-1)))
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CROSSREFS
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Cf. A000120, A000523, A008277, A035327, A053645, A054429, A059893, A290255, A329369, A341392, A357990.
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KEYWORD
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nonn,base,tabf,changed
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AUTHOR
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STATUS
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approved
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