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A065826
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Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292.
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3
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1, 1, 2, 1, 8, 3, 1, 22, 33, 4, 1, 52, 198, 104, 5, 1, 114, 906, 1208, 285, 6, 1, 240, 3573, 9664, 5955, 720, 7, 1, 494, 12879, 62476, 78095, 25758, 1729, 8, 1, 1004, 43824, 352936, 780950, 529404, 102256, 4016, 9, 1, 2026, 143520, 1820768, 6551770, 7862124, 3186344, 382720, 9117, 10
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OFFSET
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1,3
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COMMENTS
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Row sums are (n+1)!/2, i.e., A001710 offset, implying that if n balls are put at random into n boxes, the expected number of boxes with at least one ball is (n+1)/2 and the expected number of empty boxes is (n-1)/2.
T(n,k) is the number of permutations of {1,2,...,n+1} that start with an ascent and that have k-1 descents. - Ira M. Gessel, May 02 2017
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LINKS
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FORMULA
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T(n, k) = k*(a(n-1, k) + a(n-1, k-1)*(n-k+1)/(k-1)) [with T(n, 1) = 1] = Sum_{j=0..k} k*(-1)^j*(k-j)^n*binomial(n+1, j).
E.g.f.: (exp(x*(1-t)) - 1 - x*(1-t))/((1-t)*(1 - t*exp(x*(1-t)))). - Ira M. Gessel, May 02 2017
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EXAMPLE
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Rows start:
1;
1, 2;
1, 8, 3;
1, 22, 33, 4;
1, 52, 198, 104, 5;
1, 114, 906, 1208, 285, 6;
1, 240, 3573, 9664, 5955, 720, 7;
1, 494, 12879, 62476, 78095, 25758, 1729, 8;
etc.
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MAPLE
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T:=(n, k)->add(k*(-1)^j*(k-j)^n*binomial(n+1, j), j=0..k): seq(seq(T(n, k), k=1..n), n=1..10); # Muniru A Asiru, Mar 09 2019
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MATHEMATICA
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PROG
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(GAP) Flat(List([1..10], n->List([1..n], k->Sum([0..k], j->k*(-1)^j*(k-j)^n*Binomial(n+1, j))))); # Muniru A Asiru, Mar 09 2019
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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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