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 A098742 Number of indecomposable set partitions of [1..n] without singletons. 4
 0, 0, 1, 1, 3, 9, 33, 135, 609, 2985, 15747, 88761, 531561, 3366567, 22462017, 157363329, 1154257683, 8841865833, 70573741857, 585753925047, 5046128460801, 45044554041897, 416005748766771, 3969321053484921, 39077616720410409 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS After a(3) = 1, always divisible by 3. a(n) is 3 times a prime (A001748) when n = 5, 6, 11, 14, 15, 16, 19. - Jonathan Vos Post, Jun 22 2008 REFERENCES D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.7, Problem 26. George Puttenham, The Arte of English Poesie (1589), page 72, can be said to have stated the problem; but he omitted one case for n=5 and 22 cases for n=6, so he must have had other constraints in mind! LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 FORMULA If f(z) is the generating function for A074664, then a(z)=zf(z)/(1+z-f(z)). Also, if g(z) is the generating function for A000296, then a(z) = 1-1/g(z). O.g.f.: x^2/(1-x-2*x^2/(1-2*x-3*x^2/(1-3*x-4*x^2/(1-... -n*x-(n+1)*x^2/(1- ...)))))) (continued fraction). - Paul D. Hanna, Jan 17 2006 From Sergei N. Gladkovskii, Sep 20 2012, Nov 04 2012, Feb 04 2013, Feb 23 2013, Apr 18 2013, May 12 2013: (Start) Continued fractions: G.f.: -x + 2*x/E(0) where E(k)= 1 + 1/(1 + 2*x/(1 - 2*(k+2)*x/E(k+1))). G.f.: 1 - x*U(0,1/x) where U(k,x)= x - k - (k+1)/U(k+1,x). G.f.: (1+x)*x/G(0) - x where G(k) = 1 + x - x*(k+1)/(1 - x/G(k+1)). G.f.: x/Q(0) - x where Q(k)= 1 + x/(x*k-x-1)/Q(k+1). G.f.: 1 - Q(0) where Q(k)= 1 + x - x/(1 - x*(k+1)/Q(k+1)). G.f.: 1-x-1/Q(0) where Q(k)= 1 + x/(1 - x - x*(k+1)/(x + 1/Q(k+1))). (End) EXAMPLE a(5)=9 because of the set partitions 135|24, 134|25, 125|34, 145|23, 15|234, 13|245, 124|35, 12345, 14|235. [Puttenham missed the last of these.] MAPLE F:= proc(n) option remember; convert(series(1 -1/add(coeff(series(exp(exp(x)-1), x, n+1), x, j)*j!*x^j, j=0..n), x, n+1), polynom) end: a:= n-> coeff(series(x*F(n)/(1+x-F(n)), x, n+1), x, n): seq(a(n), n=0..24); # Alois P. Heinz, Sep 05 2008 MATHEMATICA f[n_] := f[n] = Normal[ Series[ 1-1/Sum[ SeriesCoefficient[ Series[ Exp[Exp[x] - 1], {x, 0, n + 1}], {x, 0, j}]*j!*x^j, {j, 0, n}], {x, 0, n + 1}]]; a[0] = 0; a[n_] := SeriesCoefficient[ Series[ x*(f[n]/(1 + x - f[n])), {x, 0, n + 1}], {x, 0, n}]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, translated from Maple *) PROG (Sage) def A098742_list(dim):     T = matrix(ZZ, dim, dim)     for n in range(dim): T[n, n] = 1     for n in (1..dim-1):         for k in (0..n-1):             T[n, k] = T[n-1, k-1]+(k+1)*T[n-1, k]+(k+2)*T[n-1, k+1]     return [0, 0]+list(T.column(0)) A098742_list(23) # - Peter Luschny, Sep 20 2012 CROSSREFS Cf. A000296, A074664, A001748, A008585. Sequence in context: A217617 A320181 A238113 * A320182 A320183 A320184 Adjacent sequences:  A098739 A098740 A098741 * A098743 A098744 A098745 KEYWORD nice,nonn,easy AUTHOR Don Knuth, Oct 01 2004 EXTENSIONS More terms from Vladeta Jovovic, Oct 21 2004 STATUS approved

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Last modified January 20 08:12 EST 2021. Contains 340301 sequences. (Running on oeis4.)