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A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.

%I A086885 #41 Feb 16 2025 08:32:50
%S A086885 2,3,7,4,13,34,5,21,73,209,6,31,136,501,1546,7,43,229,1045,4051,13327,
%T A086885 8,57,358,1961,9276,37633,130922,9,73,529,3393,19081,93289,394353,
%U A086885 1441729,10,91,748,5509,36046,207775,1047376,4596553,17572114,11,111,1021,8501
%N A086885 Lower triangular matrix, read by rows: T(i,j) = number of ways i seats can be occupied by any number k (0<=k<=j<=i) of persons.
%C A086885 Compare with A088699. - _Peter Bala_, Sep 17 2008
%C A086885 T(m, n) gives the number of matchings in the complete bipartite graph K_{m,n}. - _Eric W. Weisstein_, Apr 25 2017
%H A086885 Robert Israel, <a href="/A086885/b086885.txt">Table of n, a(n) for n = 1..10011</a> (rows 1 to 141, flattened)
%H A086885 Ed Jones, <a href="https://groups.google.com/group/sci.math/msg/590026edeca21c52">Number of seatings</a>, discussion in newsgroup sci.math, Aug 9, 2003.
%H A086885 R. J. Mathar, <a href="/A247158/a247158.pdf">The number of binary nxm matrices with at most k 1's in each row or columns</a>, Table 1.
%H A086885 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A086885 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>
%H A086885 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching.html">Matching</a>
%H A086885 Luca Zecchini, Tobias Bleifuß, Giovanni Simonini, Sonia Bergamaschi, and Felix Naumann, <a href="https://doi.org/10.1145/3639303">Determining the Largest Overlap between Tables</a>, Proc. ACM Manag. Data (SIGMOD 2024) Vol. 2, No. 1, Art. 48. See p. 48:6.
%H A086885 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A086885 a(n) = T(i, j) with n=(i*(i-1))/2+j; T(i, 1)=i+1, T(i, j)=T(i, j-1)+i*T(i-1, j-1) for j>1.
%F A086885 The role of seats and persons may be interchanged, so T(i, j)=T(j, i).
%F A086885 T(i, j) = j!*LaguerreL(j, i-j, -1). - _Vladeta Jovovic_, Aug 25 2003
%F A086885 T(i, j) = Sum_{k=0..j} k!*binomial(i, k)*binomial(j, k). - _Vladeta Jovovic_, Aug 25 2003
%e A086885 One person:
%e A086885 T(1,1)=a(1)=2: 0,1 (seat empty or occupied);
%e A086885 T(2,1)=a(2)=3: 00,10,01 (both seats empty, left seat occupied, right seat occupied).
%e A086885 Two persons:
%e A086885 T(2,2)=a(3)=7: 00,10,01,20,02,12,21;
%e A086885 T(3,2)=a(5)=13: 000,100,010,001,200,020,002,120,102,012,210,201,021.
%e A086885 Triangle starts:
%e A086885   2;
%e A086885   3  7;
%e A086885   4 13  34;
%e A086885   5 21  73 209;
%e A086885   6 31 136 501 1546;
%e A086885   ...
%p A086885 A086885 := proc(n,k)
%p A086885     add( binomial(n,j)*binomial(k,j)*j!,j=0..min(n,k)) ;
%p A086885 end proc: # _R. J. Mathar_, Dec 19 2014
%t A086885 Table[Table[Sum[k! Binomial[n, k] Binomial[j, k], {k, 0, j}], {j, 1, n}], {n, 1, 10}] // Grid (* _Geoffrey Critzer_, Jul 09 2015 *)
%t A086885 Table[m! LaguerreL[m, n - m, -1], {n, 10}, {m, n}] // Flatten (* _Eric W. Weisstein_, Apr 25 2017 *)
%o A086885 (Sage) flatten([[factorial(k)*gen_laguerre(k, n-k, -1) for k in [1..n]] for n in (1..10)]) # _G. C. Greubel_, Feb 23 2021
%o A086885 (Magma) [Factorial(k)*Evaluate(LaguerrePolynomial(k, n-k), -1): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Feb 23 2021
%o A086885 (PARI) T(i, j) = j!*pollaguerre(j, i-j, -1); \\ _Michel Marcus_, Feb 23 2021
%Y A086885 Diagonal: A002720, first subdiagonal: A000262, 2nd subdiagonal: A052852, 3rd subdiagonal: A062147, 4th subdiagonal: A062266, 5th subdiagonal: A062192, 2nd row/column: A002061. With column 0: A176120.
%K A086885 nonn,easy,tabl
%O A086885 1,1
%A A086885 _Hugo Pfoertner_, Aug 22 2003