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 A005490 Number of partitions of [n] where the first k elements are marked (0 <= k <= n-1) and at least k blocks contain their own index. (Formerly M3467) 4
 1, 4, 13, 44, 163, 666, 2985, 14550, 76497, 430746, 2582447, 16403028, 109918745, 774289168, 5715471605, 44087879136, 354521950931, 2965359744446, 25749723493073, 231719153184018, 2157494726318233, 20753996174222510, 205985762120971167, 2106795754056142536 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Old name was: From expansion of falling factorials. REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Table of n, a(n) for n=1..24. E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317. FORMULA a(n) = Sum_{i=1..n} b(n, i) where b(n, 1) = n and b(n+1, i+1) = (n-i) * b(n, i) + b(n+1, i) [From Whitehead]. - Sean A. Irvine, Jul 01 2016 From Alois P. Heinz, Jan 05 2022: (Start) a(n) = Sum_{k=0..n-1} A108087(n-k,k). a(n) = A000110(n) + Sum_{k=1..n-1} A259691(n,k)/k. a(n) = A347420(n) - 1. a(n) mod 2 = n mod 2 = A000035(n). (End) EXAMPLE a(3) = 13 = 5 + 5 + 3: 123, 12|3, 13|2, 1|23, 1|2|3, 1'23, 1'2|3, 1'3|2, 1'|23, 1'|2|3, 1'3|2', 1'|2'3, 1'|2'|3. MAPLE b:= proc(n, m) option remember; `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m)) end: a:= n-> add(b(n-k, k), k=0..n-1): seq(a(n), n=1..24); # Alois P. Heinz, Jan 05 2022 MATHEMATICA b[n_, m_] := b[n, m] = If[n == 0, 1, b[n - 1, m + 1] + m*b[n - 1, m]]; a[n_] := Sum[b[n - k, k], {k, 0, n - 1}]; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Apr 24 2022, after Alois P. Heinz *) CROSSREFS Cf. A000035, A000110, A108087, A259691, A347420. Sequence in context: A149430 A124463 A081197 * A174299 A351028 A285188 Adjacent sequences: A005487 A005488 A005489 * A005491 A005492 A005493 KEYWORD nonn AUTHOR N. J. A. Sloane, Simon Plouffe EXTENSIONS More terms from Sean A. Irvine, Jul 01 2016 New name from Alois P. Heinz, Jan 07 2022 STATUS approved

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Last modified June 8 17:45 EDT 2023. Contains 363165 sequences. (Running on oeis4.)