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A144108 Eigentriangle based on A052186 (permutations without strong fixed points), row sums = n!

%I A144108 #5 Oct 30 2012 00:54:59
%S A144108 1,0,1,1,0,1,3,1,0,2,14,3,1,0,6,77,14,3,2,0,24,497,77,14,6,6,0,120,
%T A144108 3676,497,77,28,18,24,0,720,30677,3676,497,154,84,72,120,0,5040,
%U A144108 285335,30677,3676,994,462,336,360,720,0,40320
%N A144108 Eigentriangle based on A052186 (permutations without strong fixed points), row sums = n!
%C A144108 Row sums = n!. Sum n-th row terms = rightmost term of next row.
%C A144108 Left border = A052186.
%F A144108 Eigentriangle by rows, T(n,k) = A052186(n-k)*X; 0<=k<=n; where X = an infinite lower triangular matrix with the factorials shifted to (1, 1, 1, 2, 6, 24,...) in the main diagonal and the rest zeros. The triangle A052186 is composed of A052186 in every column: (1, 0, 1, 3, 14, 77, 497,...). The operations are equivalent to (by rows): termwise products of (n+1) terms of A052186 (reversed) and an equal number of terms in the series: (1, 1, 1, 2, 6, 24, 120,...).
%e A144108 First few rows of the triangle =
%e A144108 1;
%e A144108 0, 1;
%e A144108 1, 0, 1;
%e A144108 3, 1, 0, 2;
%e A144108 14, 3, 1, 0, 6;
%e A144108 77, 14, 3, 2, 0, 24;
%e A144108 497, 77, 14, 6, 6, 0, 120;
%e A144108 3676, 497, 77, 28, 18, 24, 0, 720;
%e A144108 30677, 3676, 497, 154, 84, 72, 120, 0, 5040;
%e A144108 285335, 30677, 3676, 994, 462, 336, 360, 720, 0, 40320;
%e A144108 ...
%e A144108 Row 3 = (14, 3, 1, 0, 6) = termwise products of (14, 3, 1, 0, 1) and (1, 1, 1, 2, 6) = (14*1, 3*1, 1*1, 0*2, 1*6).
%Y A144108 A000142, Cf. A052186
%K A144108 nonn,tabl
%O A144108 0,7
%A A144108 _Gary W. Adamson_, Sep 11 2008
%E A144108 Typo in row for n=7 corrected by _Olivier Gérard_, Oct 30 2012