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A202191 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).

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