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 A264902 Number T(n,k) of defective parking functions of length n and defect k; triangle T(n,k), n>=0, 0<=k<=max(0,n-1), read by rows. 17
 1, 1, 3, 1, 16, 10, 1, 125, 107, 23, 1, 1296, 1346, 436, 46, 1, 16807, 19917, 8402, 1442, 87, 1, 262144, 341986, 173860, 41070, 4320, 162, 1, 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1, 100000000, 148717762, 96920092, 34268902, 6768184, 710314, 34660, 574, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Rows n = 0..141, flattened Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008 FORMULA T(n,k) = S(n,k) - S(n,k+1) with S(n,0) = n^n, S(n,k) = Sum_{i=0..n-k} C(n,i) * k*(k+i)^(i-1) * (n-k-i)^(n-i) for k>0. Sum_{k>0} k * T(n,k) = A036276(n-1) for n>0. Sum_{k>0} T(n,k) = A101334(n). Sum_{k>=0} (-1)^k * T(n,k) = A274279(n) for n>=1. EXAMPLE T(2,0) = 3: [1,1], [1,2], [2,1]. T(2,1) = 1: [2,2]. T(3,1) = 10: [1,3,3], [2,2,2], [2,2,3], [2,3,2], [2,3,3], [3,1,3], [3,2,2], [3,2,3], [3,3,1], [3,3,2]. T(3,2) = 1: [3,3,3]. Triangle T(n,k) begins: 0 : 1; 1 : 1; 2 : 3, 1; 3 : 16, 10, 1; 4 : 125, 107, 23, 1; 5 : 1296, 1346, 436, 46, 1; 6 : 16807, 19917, 8402, 1442, 87, 1; 7 : 262144, 341986, 173860, 41070, 4320, 162, 1; 8 : 4782969, 6713975, 3924685, 1166083, 176843, 12357, 303, 1; ... MAPLE S:= (n, k)-> `if`(k=0, n^n, add(binomial(n, i)*k* (k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k)): T:= (n, k)-> S(n, k)-S(n, k+1): seq(seq(T(n, k), k=0..max(0, n-1)), n=0..10); MATHEMATICA S[n_, k_] := If[k==0, n^n, Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}]]; T[n_, k_] := S[n, k]-S[n, k+1]; T[0, 0] = 1; Table[T[n, k], {n, 0, 10}, {k, 0, Max[0, n-1]}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *) CROSSREFS Columns k=0-10 give: A000272(n+1), A140647, A291128, A291129, A291130, A291131, A291132, A291133, A291134, A291135, A291136. Row sums give A000312. T(2n,n) gives A264903. Cf. A036276, A101334, A274279. Sequence in context: A071211 A222029 A038675 * A350446 A156653 A048159 Adjacent sequences: A264899 A264900 A264901 * A264903 A264904 A264905 KEYWORD nonn,tabf,easy AUTHOR Alois P. Heinz, Nov 28 2015 STATUS approved

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Last modified June 16 11:09 EDT 2024. Contains 373429 sequences. (Running on oeis4.)