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 A253409 The number of ways to write an n-bit binary string and then define each run of ones as an element in an equivalence relation, and each run of zeros as an element in a second equivalence relation. 2
 1, 2, 4, 10, 28, 86, 282, 984, 3630, 14138, 57904, 248854, 1118554, 5246980, 25619018, 129961850, 683561488, 3722029314, 20946195078, 121671375312, 728511702462, 4491224518274, 28475638336144, 185499720543262, 1240358846060122, 8505894459387628, 59771243719783410 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Included are the cases in which there are no zeros or no ones, producing an empty relation. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..647 FORMULA a(n) = 2 * Sum_{k=1..ceiling(n/2)} C(n-1,2k-1)*Bell(k)^2 + C(n-1,2k-2)*Bell(k)*Bell(k-1), where C(x,y) refers to binomial coefficients and Bell(x) refers to Bell numbers (A000110). EXAMPLE For n = 3, taking 3-bit binary strings and replacing zeros with ABC... and ones with 123... to represent equivalence relations, we have a(3) = 10 labeled-run binary strings: AAA, AA1, A1A, A1B, 1AA, A11, 11A, 111, 1A1, 1A2. MATHEMATICA Table[2 * Sum[Binomial[n-1, 2k-1] * BellB[k]^2 + Binomial[n-1, 2k-2] * BellB[k] * BellB[k-1], {k, 1, Ceiling[n/2]}], {n, 1, 30}] (* Vaclav Kotesovec, Jan 08 2015 after Andrew Woods *) CROSSREFS Cf. A000110, A247100. Sequence in context: A149829 A149830 A272484 * A027412 A030277 A149831 Adjacent sequences:  A253406 A253407 A253408 * A253410 A253411 A253412 KEYWORD nonn AUTHOR Andrew Woods, Jan 01 2015 EXTENSIONS a(0)=1 prepended by Alois P. Heinz, Aug 08 2015 STATUS approved

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Last modified June 27 19:31 EDT 2022. Contains 354898 sequences. (Running on oeis4.)