cp's OEIS Frontend

A220183 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k triple descents (n>=0,0<=k<=n-3). We say that i is a triple descent of a permutation p if p(i) > p(i+1) > p(i+2) > p(i+3).

%I A220183 #33 Dec 01 2016 15:42:14
%S A220183 1,1,2,6,23,1,111,8,1,642,67,10,1,4326,602,99,12,1,33333,5742,1093,
%T A220183 137,14,1,288901,59504,12425,1852,181,16,1,2782082,666834,151635,
%U A220183 24970,3029,231,18,1,29471046,8054684,1981499,355906,48455,4902,287,20,1
%N A220183 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k triple descents (n>=0,0<=k<=n-3). We say that i is a triple descent of a permutation p if p(i) > p(i+1) > p(i+2) > p(i+3).
%C A220183 Row sums = n!.
%C A220183 T(n,0) = A117158.
%H A220183 Alois P. Heinz, <a href="/A220183/b220183.txt">Rows n = 0..143, flattened</a>
%H A220183 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 210
%F A220183 E.g.f.: 1/(1 - x - Sum_{k,n} I(n,k)(y - 1)^k*x^n/n!) where I(n,k) is the coefficient of y^k*x^n in the ordinary generating function expansion of y x^4/(1 - y*x - y*x^2 - y*x^3) See Flajolet and Sedgewick reference in Links section.
%e A220183 :     1;
%e A220183 :     1;
%e A220183 :     2;
%e A220183 :     6;
%e A220183 :    23,    1;
%e A220183 :   111,    8,    1;
%e A220183 :   642,   67,   10,   1;
%e A220183 :  4326,  602,   99,  12,  1;
%e A220183 : 33333, 5742, 1093, 137, 14, 1;
%e A220183 T(5,1) = 8 because we have: (4,5,3,2,1), (3,5,4,2,1), (2,5,4,3,1), (5,4,3,1,2), (1,5,4,3,2), (5,4,2,1,3), (5,3,2,1,4), (4,3,2,1,5).
%p A220183 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
%p A220183       add(b(u-j, o+j-1, 1), j=1..u)+
%p A220183       add(b(u+j-1, o-j, [2, 3, 3][t])*`if`(t=3, x, 1), j=1..o)))
%p A220183     end:
%p A220183 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
%p A220183 seq(T(n), n=0..14);  # _Alois P. Heinz_, Oct 29 2013
%t A220183 nn=10; u=y-1; a=Apply[Plus, Table[Normal[Series[y x^4/(1-y x - y x^2-y x^3), {x,0,nn}]][[n]]/(n+3)!, {n,1,nn-3}]]/.y->u; Range[0,nn]! CoefficientList[Series[1/(1-x-a), {x,0,nn}], {x,y}]//Grid
%Y A220183 Cf. A162975, A242783, A242784.
%K A220183 nonn,tabf
%O A220183 0,3
%A A220183 _Geoffrey Critzer_, Dec 12 2012