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 A289693 The number of partitions of [n] with exactly 3 blocks without peaks. 2
 0, 0, 1, 3, 9, 27, 75, 197, 503, 1263, 3132, 7695, 18784, 45649, 110585, 267276, 644907, 1554208, 3742321, 9005265, 21659603, 52078400, 125186565, 300870586, 723010749, 1737273406, 4174084259, 10028409724, 24092769583, 57880137331 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Table of n, a(n) for n=1..30. T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Table 1. Index entries for linear recurrences with constant coefficients, signature (6,-15,24,-29,25,-17,9,-3,1). FORMULA From Colin Barker, Nov 07 2017: (Start) G.f.: x^3*(1 - x + x^2)*(1 - 2*x + 3*x^2 - x^3 + x^4) / ((1 - x)*(1 - 2*x + x^2 - x^3)*(1 - 3*x + 3*x^2 - 4*x^3 + x^4 - x^5)). a(n) = 6*a(n-1) - 15*a(n-2) + 24*a(n-3) - 29*a(n-4) + 25*a(n-5) - 17*a(n-6) + 9*a(n-7) - 3*a(n-8) + a(n-9) for n>9. (End) MAPLE with(orthopoly) : nmax := 10: tpr := 1+x^2/2 : k := 3: g := x^k ; for j from 1 to k do if j> 1 then g := g*( U(j-1, tpr)-(1+x)*U(j-2, tpr)) / ((1-x)*U(j-1, tpr)-U(j-2, tpr)) ; else # note that U(-1, .)=0, U(0, .)=1 g := g* U(j-1, tpr) / ((1-x)*U(j-1, tpr)) ; end if; end do: simplify(%) ; taylor(g, x=0, nmax+1) ; gfun[seriestolist](%) ; # R. J. Mathar, Mar 11 2021 CROSSREFS Cf. A289692, A289694. Sequence in context: A193703 A289658 A180238 * A269684 A330079 A361423 Adjacent sequences: A289690 A289691 A289692 * A289694 A289695 A289696 KEYWORD nonn AUTHOR R. J. Mathar, Jul 09 2017 STATUS approved

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Last modified December 6 18:41 EST 2023. Contains 367614 sequences. (Running on oeis4.)