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%I #39 Mar 07 2019 10:03:33
%S 0,0,0,0,22,172,856,3488,12746,43628,143244,457536,1434318,4438540,
%T 13611136,41473216,125797010,380341580,1147318004,3455325600,
%U 10394291094,31242645420,93853769320,281825553760,846030314842,2539248578732,7620161662556,22865518160768
%N Number of permutations on [n] whose up-down signature has nonnegative partial sums and which have exactly two descents.
%C Also the number of permutations of [n] of odd order whose M statistic (as defined in the Spiro paper) is equal to two.
%H Sam Spiro, <a href="/A321268/b321268.txt">Table of n, a(n) for n = 1..100</a>
%H S. Spiro, <a href="https://arxiv.org/abs/1810.00993">Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic</a>, arXiv:1810.00993 [math.CO], 2018.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (11,-50,122,-173,143,-64,12).
%F a(n) = 3*A008292(n-1,3)- 2*binomial(n,3)+binomial(n,2)-1 for n > 1.
%F a(n) = A065826(n-1,3)- 2*binomial(n,3)+binomial(n,2)-1 for n > 1.
%F a(n) = 3^n-3*n*2^(n-1)-2*binomial(n,3)+4*binomial(n,2)-1 for n > 1.
%F From _Colin Barker_, Mar 07 2019: (Start)
%F G.f.: 2*x^5*(11 - 35*x + 32*x^2 - 6*x^3) / ((1 - x)^4*(1 - 2*x)^2*(1 - 3*x)).
%F a(n) = 11*a(n-1) - 50*a(n-2) + 122*a(n-3) - 173*a(n-4) + 143*a(n-5) - 64*a(n-6) + 12*a(n-7) for n>8.
%F a(n) = -1 + 3^n - (16+9*2^n)*n/6 + 3*n^2 - n^3/3 for n>1.
%F (End)
%e Some permutations counted by a(5) include 14253 and 34521.
%t a[1] = 0; a[n_] := 2n^2 - 2n - 1 - n 2^(n-1) - 2 Binomial[n, 3] + Sum[ Binomial[n, k] (2^k - 2k), {k, 0, n}];
%t Table[a[n], {n, 1, 28}] (* _Jean-François Alcover_, Nov 11 2018 *)
%o (PARI) a(n)={if(n<2, 0, 2*n^2 - 2*n - 1 - n*2^(n-1) - 2*binomial(n,3) + sum(k=0, n, binomial(n, k)*(2^k - 2*k)))} \\ _Andrew Howroyd_, Nov 01 2018
%o (PARI) concat([0,0,0,0], Vec(2*x^5*(11 - 35*x + 32*x^2 - 6*x^3) / ((1 - x)^4*(1 - 2*x)^2*(1 - 3*x)) + O(x^40))) \\ _Colin Barker_, Mar 07 2019
%Y Cf. A000246, A005803, A321269, A303285, A177042.
%Y Column k=2 of A321280.
%K nonn,easy
%O 1,5
%A _Sam Spiro_, Nov 01 2018
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