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Irregular triangle read by rows: row n gives expansion of (1-x+x^2)*(1+x)^n.
2

%I #33 Mar 20 2023 10:42:06

%S 1,-1,1,1,0,0,1,1,1,0,1,1,1,2,1,1,2,1,1,3,3,2,3,3,1,1,4,6,5,5,6,4,1,1,

%T 5,10,11,10,11,10,5,1,1,6,15,21,21,21,21,15,6,1,1,7,21,36,42,42,42,36,

%U 21,7,1,1,8,28,57,78,84,84,78,57,28,8,1,1,9,36,85,135,162,168,162,135,85,36,9,1

%N Irregular triangle read by rows: row n gives expansion of (1-x+x^2)*(1+x)^n.

%F G.f.: A(x,y) = (1 - y + y^2)/(1 - x*(1+y)). - _Andrew Howroyd_, Mar 03 2023

%e Row n=0 : 1, -1, 1 ;

%e Row n=1 : 1, 0, 0, 1 ;

%e Row n=2 : 1, 1, 0, 1, 1 ;

%e Row n=3 : 1, 2, 1, 1, 2, 1 ;

%e Row n=4 : 1, 3, 3, 2, 3, 3, 1 ;

%e Row n=5 : 1, 5, 10, 11, 10, 11, 10, 5, 1;

%e Row n=6 : 1, 6, 15, 21, 21, 21, 21, 15, 6, 1;

%e ...

%o (PARI) row(n)=Vecrev(polcoef((1 - y + y^2)/(1 - x*(1+y)) + O(x*x^n), n))

%o { for(n=0, 10, print(row(n))) } \\ _Andrew Howroyd_, Mar 03 2023

%Y Cf. A000079, A014626, A024483, A050407

%K sign,tabf

%O 0,14

%A _Philippe Deléham_, Apr 07 2009