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 A008291 Triangle of rencontres numbers. 24
 1, 2, 3, 9, 8, 6, 44, 45, 20, 10, 265, 264, 135, 40, 15, 1854, 1855, 924, 315, 70, 21, 14833, 14832, 7420, 2464, 630, 112, 28, 133496, 133497, 66744, 22260, 5544, 1134, 168, 36, 1334961, 1334960, 667485, 222480, 55650, 11088, 1890, 240, 45, 14684570 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS T(n,k) = number of permutations of n elements with k fixed points. T(n,n-1)=0 and T(n,n)=1 are omitted from the array. - Geoffrey Critzer, Nov 28 2011. REFERENCES R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194. Kaufmann, Arnold. "Introduction a la combinatorique en vue des applications." Dunod, Paris, 1968. See p. 92. J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65. LINKS T. D. Noe, Rows n=2..50, flattened FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914. FORMULA T(n,k) = binomial(n,k)*A000166(n-k) = A008290(n,k). E.g.f. for column k: (x^k/k!)(exp(-x)/(1-x)). - Geoffrey Critzer, Nov 28 2011 Row generating polynomials appear to be given by -1 + sum {k = 0..n} (-1)^(n+k)*C(n,k)*(1+k*x)^(n-k)*(2+(k-1)*x)^k. - Peter Bala, Dec 29 2011 EXAMPLE Triangle begins:        1        2      3        9      8     6       44     45    20    10      265    264   135    40   15     1854   1855   924   315   70   21    14833  14832  7420  2464  630  112  28   133496 133497 66744 22260 5544 1134 168 36 MAPLE T:= proc(n, k) T(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*       (T(n-1, 0)+T(n-2, 0))), binomial(n, k)*T(n-k, 0))     end: seq(seq(T(n, k), k=0..n-2), n=2..12);  # Alois P. Heinz, Mar 17 2013 MATHEMATICA Prepend[Flatten[f[list_]:=Select[list, #>1&]; Map[f, Drop[Transpose[Table[d = Exp[-x]/(1 - x); Range[0, 10]! CoefficientList[Series[d x^k/k!, {x, 0, 10}], x], {k, 0, 8}]], 3]]], 1] (* Geoffrey Critzer, Nov 28 2011 *) PROG (PARI) T(n, k)= if(k<0 || k>n, 0, n!/k!*sum(i=0, n-k, (-1)^i/i!)) CROSSREFS Cf. A008290, A170942. Diagonals give A000217, A007290, A060008, A060836, A000166, A000240, A000387, A000449, A000475. Cf. A320582. Sequence in context: A152812 A246825 A086565 * A261525 A122665 A133066 Adjacent sequences:  A008288 A008289 A008290 * A008292 A008293 A008294 KEYWORD nonn,tabl,nice,easy AUTHOR EXTENSIONS Comments and more terms from Michael Somos, Apr 26 2000 STATUS approved

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Last modified May 30 08:43 EDT 2020. Contains 334712 sequences. (Running on oeis4.)