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A182319 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. 1
1, 1, 1, 2, 4, 1, 7, 16, 9, 1, 33, 83, 64, 16, 1, 197, 530, 486, 180, 25, 1, 1419, 4026, 4144, 1930, 410, 36, 1, 11966, 35462, 39746, 21574, 5965, 812, 49, 1, 115575, 355368, 425762, 258426, 85589, 15477, 1456, 64, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

This triangle is a refinement of A179525.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

FORMULA

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.

T(n+1,1) = sum {k = 1..n} T(n,k); T(n+1,n) = n^2.

First column A179525. Row sums A179525.

EXAMPLE

Triangle begins

.n\k.|...1.....2.....3.....4.....5

= = = = = = = = = = = = = = = = = =

..1..|...1

..2..|...1.....1

..3..|...2.....4.....1

..4..|...7....16.....9.....1

..5..|..33....83....64....16.....1

..6..|.197...530...486...180....25.....1

...

T(3,2) = 4: there is one 2x2 matrix and three 3x3 matrices with the specified properties:

........1..0..0.....0..1..0.....0..0..1..

1.1.....0..0..1.....0..1..0.....0..1..0..

0.1.....0..0..1.....0..0..1.....0..0..1..

PROG

(PARI) \\ A(n) returns vector of n'th row.

A(n)={ my(rv=if(n==1, [1], vector(n)));

my(M=matrix(2, n, k, s, k==2&&s==1));

\\ M[k, s] is number of configs with s 1's with k+1 on diagonal.

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]))));

  for(i=1, r, rv[i]+=M[i+1, n])); rv }

for(n=1, 10, print(A(n))); \\ Andrew Howroyd, Oct 10 2017

CROSSREFS

A179525.

Sequence in context: A112852 A121531 A127554 * A204182 A103324 A221073

Adjacent sequences:  A182316 A182317 A182318 * A182320 A182321 A182322

KEYWORD

nonn,tabl

AUTHOR

Peter Bala, Apr 24 2012

EXTENSIONS

Terms a(23) and beyond from Andrew Howroyd, Oct 10 2017

STATUS

approved

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Last modified October 19 22:00 EDT 2018. Contains 316378 sequences. (Running on oeis4.)