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%I #15 Oct 09 2022 09:25:53
%S 1,1,1,1,2,1,2,3,3,1,3,6,6,4,1,4,11,13,10,5,1,6,18,27,24,15,6,1,9,30,
%T 51,55,40,21,7,1,13,50,94,116,100,62,28,8,1,19,81,171,234,231,168,91,
%U 36,9,1,28,130,303,460,505,420,266,128,45,10,1,41,208
%N Triangle T(n,m) = coefficient of x^n in expansion of [x/(1-x-x^3)]^m = sum(n>=m, T(n,m) x^n).
%C Convolution triangle of Narayana's cows sequence A000930. - _Peter Luschny_, Oct 09 2022
%F T(n,m)=sum(k=0..n-m, binomial(k,(n-m-k)/2)*binomial(m+k-1,m-1)*((-1)^(n-m-k)+1))/2.
%e Triangle T(n, m) starts:
%e [1] 1;
%e [2] 1, 1;
%e [3] 1, 2, 1;
%e [4] 2, 3, 3, 1;
%e [5] 3, 6, 6, 4, 1;
%e [6] 4, 11, 13, 10, 5, 1;
%e [7] 6, 18, 27, 24, 15, 6, 1;
%e [8] 9, 30, 51, 55, 40, 21, 7, 1;
%e [9] 13, 50, 94, 116, 100, 62, 28, 8, 1;
%e .
%e From _R. J. Mathar_, Mar 15 2013: (Start)
%e The matrix inverse starts
%e 1;
%e -1,1;
%e 1,-2,1;
%e -2,3,-3,1;
%e 5,-6,6,-4,1;
%e -11,15,-13,10,-5,1;
%e 24,-36,33,-24,15,-6,1;
%e -57,84,-84,63,-40,21,-7,1;
%e 141,-204,208,-168,110,-62,28,-8,1.
%e (End)
%p A202191 := proc(n,k)
%p (x/(1-x-x^3))^k ;
%p coeftayl(%,x=0,n) ;
%p end proc: # _R. J. Mathar_, Mar 15 2013
%p # Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
%p PMatrix(10, n -> simplify(hypergeom([(2 - n)/3, (3 - n)/3, (1 - n)/3], [(2 - n)/2, (1 - n)/2], -27/4))); # _Peter Luschny_, Oct 09 2022
%o (Maxima)
%o T(n,m):=sum(binomial(k,(n-m-k)/2)*binomial(m+k-1,m-1)*((-1)^(n-m-k)+1),k,0,n-m)/2;
%Y Cf. A000930.
%K nonn,tabl
%O 1,5
%A _Vladimir Kruchinin_, Dec 14 2011
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