A176120 - OEIS

%I #39 Aug 11 2022 03:40:52

%S 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,

%T 1045,4051,13327,1,8,57,358,1961,9276,37633,130922,1,9,73,529,3393,

%U 19081,93289,394353,1441729,1,10,91,748,5509,36046,207775,1047376,4596553,17572114

%N Triangle read by rows: Sum_{j=0..k} binomial(n, j)*binomial(k, j)*j!.

%C 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

%C 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

%D O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 46.

%H Seiichi Manyama, <a href="/A176120/b176120.txt">Rows n = 0..139, flattened</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Rook_polynomial">Rook polynomial</a>

%F Sum_{k=0..n} T(n, k) = A129833(n).

%F T(n,m) = A088699(n, m). - _Peter Bala_, Aug 26 2013

%F T(n,m) = A086885(n, m). - _R. J. Mathar_, Dec 19 2014

%F From _G. C. Greubel_, Aug 11 2022: (Start)

%F T(n, k) = Hypergeometric2F1([-n, -k], [], 1).

%F T(2*n, n) = A082545(n).

%F T(2*n+1, n) = A343832(n).

%F T(n, n) = A002720(n).

%F T(n, n-1) = A000262(n), n >= 1.

%F T(n, 1) = A000027(n+1).

%F T(n, 2) = A002061(n+1).

%F T(n, 3) = A135859(n+1). (End)

%e Triangle begins

%e   1;

%e   1,  2;

%e   1,  3,   7;

%e   1,  4,  13,   34;

%e   1,  5,  21,   73,  209;

%e   1,  6,  31,  136,  501,  1546;

%e   1,  7,  43,  229, 1045,  4051,  13327;

%e   1,  8,  57,  358, 1961,  9276,  37633,  130922;

%e   1,  9,  73,  529, 3393, 19081,  93289,  394353,  1441729;

%e   1, 10,  91,  748, 5509, 36046, 207775, 1047376,  4596553, 17572114;

%e   1, 11, 111, 1021, 8501, 63591, 424051, 2501801, 12975561, 58941091, 234662231;

%p A176120 := proc(i,j)

%p         add(binomial(i,k)*binomial(j,k)*k!,k=0..j) ;

%p end proc: # _R. J. Mathar_, Jul 28 2016

%t T[n_, m_]:= T[n,m]= Sum[Binomial[n, k]*Binomial[m, k]*k!, {k, 0, m}];

%t Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

%o (Magma)

%o A176120:=func< n,k| (&+[Factorial(j)*Binomial(n,j)*Binomial(k,j): j in [0..k]]) >;

%o [A176120(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 11 2022

%o (SageMath)

%o def A176120(n,k): return sum(factorial(j)*binomial(n,j)*binomial(k,j) for j in (0..k))

%o flatten([[A176120(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Aug 11 2022

%Y Cf. A086885 (table without column 0), A129833 (row sums).

%Y Cf. A000262, A002061, A002720, A082545, A135859, A343832.

%K nonn,easy,tabl

%O 0,3

%A _Roger L. Bagula_, Apr 09 2010