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A176120
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Triangle read by rows: Sum_{j=0..k} binomial(n, j)*binomial(k, j)*j!.
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5
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1, 1, 2, 1, 3, 7, 1, 4, 13, 34, 1, 5, 21, 73, 209, 1, 6, 31, 136, 501, 1546, 1, 7, 43, 229, 1045, 4051, 13327, 1, 8, 57, 358, 1961, 9276, 37633, 130922, 1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 1, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114
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OFFSET
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0,3
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COMMENTS
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The number of ways of placing any number k = 0, 1, ..., min(n,m) of non-attacking rooks on an n X m chessboard. - R. J. Mathar, Dec 19 2014
Let a be a partial permutation in S the symmetric inverse semigroup on [n] with rank(a) := |image(a)| = m. Then T(n,m) = |aS| where |aS| is the size of the principal right ideal generated by a. - Geoffrey Critzer, Dec 21 2021
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REFERENCES
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O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 46.
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LINKS
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FORMULA
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T(n, k) = Hypergeometric2F1([-n, -k], [], 1).
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EXAMPLE
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Triangle begins
1;
1, 2;
1, 3, 7;
1, 4, 13, 34;
1, 5, 21, 73, 209;
1, 6, 31, 136, 501, 1546;
1, 7, 43, 229, 1045, 4051, 13327;
1, 8, 57, 358, 1961, 9276, 37633, 130922;
1, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729;
1, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114;
1, 11, 111, 1021, 8501, 63591, 424051, 2501801, 12975561, 58941091, 234662231;
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MAPLE
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add(binomial(i, k)*binomial(j, k)*k!, k=0..j) ;
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MATHEMATICA
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T[n_, m_]:= T[n, m]= Sum[Binomial[n, k]*Binomial[m, k]*k!, {k, 0, m}];
Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten
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PROG
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(Magma)
A176120:=func< n, k| (&+[Factorial(j)*Binomial(n, j)*Binomial(k, j): j in [0..k]]) >;
(SageMath)
def A176120(n, k): return sum(factorial(j)*binomial(n, j)*binomial(k, j) for j in (0..k))
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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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