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%I #69 Jan 25 2022 10:25:50
%S 1,1,5,31,233,2071,21305,249271,3270713,47580151,760192505,
%T 13234467511,249383390393,5057242311031,109820924003705,
%U 2542685745501751,62527556173577273,1627581948113854711,44708026328035782905,1292443104462527895991,39223568601129844839353
%N Stirling transform of A001563: a(0) = 1 and a(n) = Sum_{k=1..n} Stirling2(n,k)*k*k! for n >= 1.
%C The number of compatible bipartitions of a set of cardinality n for which at least one subset is not underlined. E.g., for n=2 there are 5 such bipartitions: {1 2}, {1}{2}, {2}{1}, _{1}_{2}, _{2}_{1}. A005649 is the number of bipartitions of a set of cardinality n. A000670 is the number of bipartitions of a set of cardinality n with none of the subsets underlined. - _Kyle Petersen_, Mar 31 2005
%C a(n) is the cardinality of the image set summed over "all surjections". All surjections means: onto functions f:{1, 2, ..., n} -> {1, 2, ..., k} for every k, 1 <= k <= n. a(n) = Sum_{k=1..n} A019538(n, k)*k. - _Geoffrey Critzer_, Nov 12 2012
%C From _Gus Wiseman_, Jan 15 2022: (Start)
%C For n > 1, also the number of finite sequences of length n + 1 covering an initial interval of positive integers with at least two adjacent equal parts, or non-anti-run patterns, ranked by the intersection of A348612 and A333217. The complement is counted by A005649. For example, the a(3) = 31 patterns, grouped by sum, are:
%C (1111) (1222) (1122) (1112) (1233) (1223)
%C (2122) (1221) (1121) (1332) (1322)
%C (2212) (2112) (1211) (2133) (2213)
%C (2221) (2211) (2111) (2331) (2231)
%C (1123) (3312) (3122)
%C (1132) (3321) (3221)
%C (2113)
%C (2311)
%C (3112)
%C (3211)
%C Also the number of ordered set partitions of {1,...,n + 1} with two successive vertices together in some block.
%C (End)
%H Alois P. Heinz, <a href="/A069321/b069321.txt">Table of n, a(n) for n = 0..200</a>
%H Benoit Cloitre, <a href="https://web.archive.org/web/20150418172931/http://bcmathematics.monsite-orange.fr/FractalOrderOfPrimes.pdf">On the fractal behavior of primes</a>, 2011. [internet archive]
%H Benoit Cloitre, <a href="https://www.docsity.com/pt/fractal-order-of-primes/4838318/">On the fractal behavior of primes</a>, 2011.
%H D. Foata and D. Zeilberger, <a href="http://arxiv.org/abs/math/9406220">The Graphical Major Index</a>, arXiv:math/9406220 [math.CO], 1994.
%H D. Foata and D. Zeilberger, <a href="http://dx.doi.org/10.1016/0377-0427(95)00254-5">Graphical major indices</a>, J. Comput. Appl. Math. 68 (1996), no. 1-2, 79-101.
%F Representation as an infinite series, in Maple notation: a(0) = 1 and a(n) = Sum_{k>=2} (k^n*(k-1)/(2^k))/4 for n >= 1. This is a Dobinski-type summation formula.
%F E.g.f.: (exp(x) - 1)/((2 - exp(x))^2).
%F a(n) = (1/2)*(A000670(n+1) - A000670(n)).
%F O.g.f.: 1 + Sum_{n >= 1} (2*n-1)!/(n-1)! * x^n / (Product_{k=1..n} (1 + (n + k - 1)*x)). - _Paul D. Hanna_, Oct 28 2013
%F a(n) = (A000629(n+1) - A000629(n))/4. - _Benoit Cloitre_, Oct 20 2002
%F a(n) = A232472(n-1)/2. - _Vincenzo Librandi_, Jan 03 2016
%F a(n) ~ n! * n / (4 * (log(2))^(n+2)). - _Vaclav Kotesovec_, Jul 01 2018
%F a(n > 0) = A000607(n + 1) - A005649(n). - _Gus Wiseman_, Jan 15 2022
%p b:= proc(n) option remember; `if`(n=0, 1,
%p add(b(n-j)*binomial(n, j), j=1..n))
%p end:
%p a:= n-> `if`(n=0, 2, b(n+1)-b(n))/2:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Feb 02 2018
%t max = 20; t = Sum[n^(n - 1)x^n/n!, {n, 1, max}]; Range[0, max]!CoefficientList[Series[D[1/(1 - y(Exp[x] - 1)), y] /. y -> 1, {x, 0, max}], x] (* _Geoffrey Critzer_, Nov 12 2012 *)
%t Prepend[Table[Sum[StirlingS2[n, k]*k*k!, {k, n}], {n, 18}], 1] (* _Michael De Vlieger_, Jan 03 2016 *)
%t a[n_] := (PolyLog[-n-1, 1/2] - PolyLog[-n, 1/2])/4; a[0] = 1; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 30 2016 *)
%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t Table[Length[Select[Join@@Permutations/@allnorm[n],MemberQ[Differences[#],0]&]],{n,0,8}] (* _Gus Wiseman_, Jan 15 2022 *)
%o (PARI) {a(n)=polcoeff(1+sum(m=1, n, (2*m-1)!/(m-1)!*x^m/prod(k=1, m, 1+(m+k-1)*x+x*O(x^n))), n)} \\ _Paul D. Hanna_, Oct 28 2013
%Y Cf. A000629, A001563, A232472.
%Y The complement is counted by A005649.
%Y A version for permutations of prime indices is A336107.
%Y A version for factorizations is A348616.
%Y Dominated (n > 1) by A350252, complement A345194, compositions A345192.
%Y A000670 = patterns, ranked by A333217.
%Y A001250 = alternating permutations, complement A348615.
%Y A003242 = anti-run compositions, ranked by A333489.
%Y A019536 = necklace patterns.
%Y A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
%Y A261983 = not-anti-run compositions, ranked by A348612.
%Y A333381 = anti-runs of standard compositions.
%Y Cf. A000110, A008277, A055932, A106356, A238424, A333382, A335456, A336103, A349050, A349055, A350138.
%K nonn
%O 0,3
%A _Karol A. Penson_, Mar 14 2002
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