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 A097592 Triangle read by rows: T(n,k) is the number of permutations of [n] with exactly k increasing runs of even length. 18
 1, 1, 1, 1, 2, 4, 7, 12, 5, 25, 52, 43, 102, 299, 258, 61, 531, 1750, 1853, 906, 3141, 11195, 15634, 8965, 1385, 20218, 83074, 133697, 94398, 31493, 146215, 675304, 1207256, 1088575, 460929, 50521, 1174889, 5880354, 11974457, 12625694, 6632158 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row n has 1+floor(n/2) entries. LINKS Alois P. Heinz, Rows n = 0..180, flattened FORMULA E.g.f.: 2(t-1)u/[ -2u+(2-t+tu)exp((-1+u)x/2)+(t-2+tu)exp(-(1+u)x/2)], where u=sqrt(5-4t). Sum_{k=1..floor(n/2)} k * T(n,k) = A097593(n). - Alois P. Heinz, Jul 04 2019 EXAMPLE Triangle starts:     1;     1;     1,   1;     2,   4;     7,  12,   5;    25,  52,  43;   102, 299, 258, 61; Example: T(4,2) = 5 because we have 13/24, 14/23, 23/14, 24/13 and 34/12. MAPLE G:=2*(t-1)*u/(-2*u+(2-t+t*u)*exp((-1+u)*x/2)+(t-2+t*u)*exp(-(1+u)*x/2)): u:=sqrt(5-4*t): Gser:=simplify(series(G, x=0, 12)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(n!*coeff(Gser, x^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)), n=0..11); # second Maple program: b:= proc(u, o, t) option remember; `if`(u+o=0, x^t, expand(       add(b(u+j-1, o-j, irem(t+1, 2)), j=1..o)+       add(b(u-j, o+j-1, 0)*x^t, j=1..u)))     end: T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0\$2)): seq(T(n), n=0..12);  # Alois P. Heinz, Nov 19 2013 MATHEMATICA b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, x^t, Expand[Sum[b[u+j-1, o-j, Mod[t+1, 2]], {j, 1, o}] + Sum[b[u-j, o+j-1, 0]*x^t, {j, 1, u}]]]; T[n_] := Function[ {p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A097597, A317281, A317282, A317283, A317284, A317285, A317286, A317287, A317288, A317289, A317290. Row sums give A000142. T(n,floor(n/2)) gives A317139. T(2n,n) gives A000364. T(2n+1,n) gives A317140. Cf. A097591, A097593. Sequence in context: A153555 A259588 A058103 * A267699 A193841 A052474 Adjacent sequences:  A097589 A097590 A097591 * A097593 A097594 A097595 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Aug 29 2004 STATUS approved

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Last modified June 14 19:28 EDT 2021. Contains 345038 sequences. (Running on oeis4.)