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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. 10
 2, 3, 7, 4, 13, 34, 5, 21, 73, 209, 6, 31, 136, 501, 1546, 7, 43, 229, 1045, 4051, 13327, 8, 57, 358, 1961, 9276, 37633, 130922, 9, 73, 529, 3393, 19081, 93289, 394353, 1441729, 10, 91, 748, 5509, 36046, 207775, 1047376, 4596553, 17572114, 11, 111, 1021, 8501 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Compare with A088699. - Peter Bala, Sep 17 2008 T(m, n) gives the number of matchings in the complete bipartite graph K_{m,n}. - Eric W. Weisstein, Apr 25 2017 LINKS Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened) Ed Jones, Number of seatings, discussion in newsgroup sci.math, Aug 9, 2003. R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or columns, Table 1. Eric Weisstein's World of Mathematics, Complete Bipartite Graph Eric Weisstein's World of Mathematics, Independent Edge Set Eric Weisstein's World of Mathematics, Matching FORMULA 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 The role of seats and persons may be interchanged, so T(i, j)=T(j, i). T(i, j) = j!*LaguerreL(j, i-j, -1). - Vladeta Jovovic, Aug 25 2003 T(i, j) = Sum_{k=0..j} k!*binomial(i, k)*binomial(j, k). - Vladeta Jovovic, Aug 25 2003 EXAMPLE One person: T(1,1)=a(1)=2: 0,1 (seat empty or occupied); T(2,1)=a(2)=3: 00,10,01 (both seats empty, left seat occupied, right seat occupied). Two persons: T(2,2)=a(3)=7: 00,10,01,20,02,12,21; T(3,2)=a(5)=13: 000,100,010,001,200,020,002,120,102,012,210,201,021. Triangle starts:   2;   3  7;   4 13  34;   5 21  73 209;   6 31 136 501 1546;   ... MAPLE A086885 := proc(n, k)     add( binomial(n, j)*binomial(k, j)*j!, j=0..min(n, k)) ; end proc: # R. J. Mathar, Dec 19 2014 MATHEMATICA 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 *) Table[m! LaguerreL[m, n - m, -1], {n, 10}, {m, n}] // Flatten (* Eric W. Weisstein, Apr 25 2017 *) PROG (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 (Magma) [Factorial(k)*Evaluate(LaguerrePolynomial(k, n-k), -1): k in [1..n], n in [1..10]]; // G. C. Greubel, Feb 23 2021 (PARI) T(i, j) = j!*pollaguerre(j, i-j, -1); \\ Michel Marcus, Feb 23 2021 CROSSREFS 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. Sequence in context: A287628 A319863 A320948 * A324598 A229794 A331318 Adjacent sequences:  A086882 A086883 A086884 * A086886 A086887 A086888 KEYWORD nonn,easy,tabl AUTHOR Hugo Pfoertner, Aug 22 2003 STATUS approved

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Last modified July 4 11:31 EDT 2022. Contains 355075 sequences. (Running on oeis4.)