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%I #13 Oct 10 2017 15:36:15
%S 1,1,1,2,4,1,7,16,9,1,33,83,64,16,1,197,530,486,180,25,1,1419,4026,
%T 4144,1930,410,36,1,11966,35462,39746,21574,5965,812,49,1,115575,
%U 355368,425762,258426,85589,15477,1456,64,1
%N Triangular array: T(n,k) counts upper triangular matrices with entries from {0,1} having n 1's in total, with k 1's on the main diagonal and at least one nonzero entry in each row.
%C This triangle is a refinement of A179525.
%H Andrew Howroyd, <a href="/A182319/b182319.txt">Table of n, a(n) for n = 1..1275</a>
%F Let F(x,t) = 1 - (1-t)*(1 - (1-x*t) + (1-x*t)*(1-x^2*t) - (1-x*t)*(1-x^2*t)*(1-x^3*t) + ...). Then F(1+x,1+x*t) = 1 + x*t + (t+t^2)*x^2 + (2*t+4*t^2+t^3)*x^3 + (7*t+16*t^2+9*t^3+t^4)*x^4 + ... is conjecturally a generating function for the triangle.
%F T(n+1,1) = sum {k = 1..n} T(n,k); T(n+1,n) = n^2.
%F First column A179525. Row sums A179525.
%e Triangle begins
%e .n\k.|...1.....2.....3.....4.....5
%e = = = = = = = = = = = = = = = = = =
%e ..1..|...1
%e ..2..|...1.....1
%e ..3..|...2.....4.....1
%e ..4..|...7....16.....9.....1
%e ..5..|..33....83....64....16.....1
%e ..6..|.197...530...486...180....25.....1
%e ...
%e T(3,2) = 4: there is one 2x2 matrix and three 3x3 matrices with the specified properties:
%e ........1..0..0.....0..1..0.....0..0..1..
%e 1.1.....0..0..1.....0..1..0.....0..1..0..
%e 0.1.....0..0..1.....0..0..1.....0..0..1..
%o (PARI) \\ A(n) returns vector of n'th row.
%o A(n)={ my(rv=if(n==1, [1], vector(n)));
%o my(M=matrix(2,n,k,s,k==2&&s==1));
%o \\ M[k,s] is number of configs with s 1's with k+1 on diagonal.
%o for(r=2, n, M=matrix(r+1,n,k,s, sum(j=0, min(s-1,r-1), binomial(r-1,j) * (if(j>0&&k<=r, M[k,s-j]) + if(j<s-1&&k>1, M[k-1,s-j-1]))));
%o for(i=1, r, rv[i]+=M[i+1,n])); rv }
%o for(n=1,10,print(A(n))); \\ _Andrew Howroyd_, Oct 10 2017
%Y A179525.
%K nonn,tabl
%O 1,4
%A _Peter Bala_, Apr 24 2012
%E Terms a(23) and beyond from _Andrew Howroyd_, Oct 10 2017
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