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A162979 Triangle read by rows: T(n,k) is the number of alternating (i.e., down-up) permutations of {1,2,...,n} having k fixed points (n >= 0, 0 <= k <= ceiling(n/2)).

%I A162979 #16 Jul 23 2018 20:09:36
%S A162979 1,0,1,1,0,1,1,0,2,2,1,6,6,3,1,24,24,11,2,102,102,51,15,2,528,528,252,
%T A162979 68,9,2952,2952,1476,458,89,9,19008,19008,9240,2728,493,44,131112,
%U A162979 131112,65556,20868,4479,621,44,1009728,1009728,495360,152448,31182,4054,265,8271792,8271792,4135896,1334928,300954,47670,4959,265
%N A162979 Triangle read by rows: T(n,k) is the number of alternating (i.e., down-up) permutations of {1,2,...,n} having k fixed points (n >= 0, 0 <= k <= ceiling(n/2)).
%C A162979 Sum of entries in row n is the Euler (up-down) number A000111(n).
%C A162979 T(n,0) = T(n,1) = A129817(n) (n>=1).
%C A162979 T(2n,n) = T(2n+1,n+1) = d(n), where d(n) = A000166 is a derangement number (see the Chapman & Williams reference).
%C A162979 Sum_{k>=0} k*T(n,k) = A162978(n).
%H A162979 R. Chapman and L. K. Williams, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v14i1n16">A conjecture of Stanley on alternating permutations</a>, The Electronic J. of Combinatorics, 14, 2007, #N16.
%H A162979 R. P. Stanley, <a href="https://dx.doi.org/10.1016/j.jcta.2006.06.008">Alternating permutations and symmetric functions</a>, J. Comb. Theory A 114 (3) (2007) 436-460.
%F A162979 The row generating polynomials can be obtained from Proposition 6.1 of the Stanley reference (see the Maple program).
%e A162979 T(5,2)=3 because we have 32415, 41325, and 52314.
%e A162979 Triangle starts:
%e A162979     1;
%e A162979     0,   1;
%e A162979     1,   0;
%e A162979     1,   1,   0;
%e A162979     2,   2,   1;
%e A162979     6,   6,   3,   1;
%e A162979    24,  24,  11,   2;
%e A162979   102, 102,  51,  15,   2;
%p A162979 fo := exp(E*(arctan(q*t)-arctan(t)))/(1-E*t): fe := sqrt((1+t^2)/(1+q^2*t^2))*exp(E*(arctan(q*t)-arctan(t)))/(1-E*t): foser := simplify(series(fo, t = 0, 18)): feser := simplify(series(fe, t = 0, 18)): Q := proc (n) if `mod`(n, 2) = 1 then coeff(foser, t, n) else coeff(feser, t, n) end if end proc: for n from 0 to 16 do Q(n) end do: g := sec(x)+tan(x): gser := series(g, x = 0, 20): for n from 0 to 18 do a[n] := factorial(n)*coeff(gser, x, n) end do: for n from 0 to 15 do P[n] := sort(subs({E^14 = a[14], E^15 = a[15], E^16 = a[16], E = a[1], E^2 = a[2], E^3 = a[3], E^4 = a[4], E^5 = a[5], E^6 = a[6], E^7 = a[7], E^8 = a[8], E^9 = a[9], E^10 = a[10], E^11 = a[11], E^12 = a[12], E^13 = a[13]}, Q(n))) end do: for n from 0 to 13 do seq(coeff(P[n], q, j), j = 0 .. ceil((1/2)*n)) end do;
%t A162979 nmax = 13;
%t A162979 fo = Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1 - e*t);
%t A162979 fe = Sqrt[(1+t^2)/(1+q^2*t^2)]*Exp[e*(ArcTan[q*t] - ArcTan[t])]/(1-e*t);
%t A162979 Q[n_] := If [OddQ[n], SeriesCoefficient[fo, {t, 0, n}],  SeriesCoefficient[fe, {t, 0, n}]] // Expand;
%t A162979 a[n_] := n!*SeriesCoefficient[Sec[x] + Tan[x], {x, 0, n}];
%t A162979 P[n_] := (Q[n] /. e^k_Integer :> a[k]) /. e :> a[1] // Expand;
%t A162979 Table[Switch[n, 0, {1}, 1, {0, 1}, 2, {1, 0}, 3, {1, 1, 0}, _, CoefficientList[P[n], q]] , {n, 0, nmax}] // Flatten (* _Jean-François Alcover_, Jul 23 2018, from Maple *)
%Y A162979 Cf. A000111, A000166, A129817, A162978, A162980.
%K A162979 nonn,tabf
%O A162979 0,9
%A A162979 _Emeric Deutsch_, Aug 06 2009