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A208956 Triangular array read by rows. T(n,k) is the number of n-permutations that have at least k fixed points with n >= 1 and 1 <= k <= n. 2
1, 1, 1, 4, 1, 1, 15, 7, 1, 1, 76, 31, 11, 1, 1, 455, 191, 56, 16, 1, 1, 3186, 1331, 407, 92, 22, 1, 1, 25487, 10655, 3235, 771, 141, 29, 1, 1, 229384, 95887, 29143, 6883, 1339, 205, 37, 1, 1, 2293839, 958879, 291394, 68914, 13264, 2176, 286, 46, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Row sums = n!

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

E.g.f. for column k: 1/(1-x) - D(x)*Sum_{i=0..k-1} x^i/i! where D(x) is the e.g.f. for A000166.

T(n,k) = Sum_{i=k..n} C(n,i)*A000166(n-i). - Alois P. Heinz, Apr 22 2013

EXAMPLE

Triangle begins:

     1;

     1,    1;

     4,    1,   1;

    15,    7,   1,  1;

    76,   31,  11,  1,  1;

   455,  191,  56, 16,  1, 1;

  3186, 1331, 407, 92, 22, 1, 1;

  ...

MAPLE

b:= proc(n) b(n):= `if`(n<2, 1-n, (n-1)*(b(n-1)+b(n-2))) end:

T:= (n, k)-> add(binomial(n, i)*b(n-i), i=k..n):

seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Apr 22 2013

MATHEMATICA

f[list_] := Select[list, #>0&]; Map[f, Transpose[Table[nn=10; d=Exp[-x]/(1-x); p=1/(1-x); s=Sum[x^i/i!, {i, 0, n}]; Drop[Range[0, nn]! CoefficientList[Series[p-s d, {x, 0, nn}], x], 1], {n, 0, 9}]]]//Flatten

CROSSREFS

Cf. A002467 (column 1), A155521 (column 2).

Sequence in context: A157013 A346876 A141724 * A271705 A320280 A343804

Adjacent sequences:  A208953 A208954 A208955 * A208957 A208958 A208959

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Mar 03 2012

STATUS

approved

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Last modified July 7 12:47 EDT 2022. Contains 355148 sequences. (Running on oeis4.)