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%I #29 Jan 21 2020 10:07:25
%S 1,0,1,0,1,2,0,1,6,6,0,1,12,34,24,0,1,20,110,210,120,0,1,30,270,974,
%T 1452,720,0,1,42,560,3248,8946,11256,5040,0,1,56,1036,8792,38338,
%U 87504,97296,40320,0,1,72,1764,20580,129834,463050,920184,930960,362880
%N Triangle T(n,k), read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.
%C Row reverse appears to be A111184. - _Peter Bala_, Feb 17 2017
%H Alois P. Heinz, <a href="/A089949/b089949.txt">Rows n = 0..140, flattened</a>
%F Sum_{k=0..n} x^(n-k)*T(n,k) = A111528(x, n); see A000142, A003319, A111529, A111530, A111531, A111532, A111533 for x = 0, 1, 2, 3, 4, 5, 6. - _Philippe Deléham_, Aug 09 2005
%F Sum_{k=0..n} T(n,k)*3^k = A107716(n). - _Philippe Deléham_, Aug 15 2005
%F Sum_{k=0..n} T(n,k)*2^k = A000698(n+1). - _Philippe Deléham_, Aug 15 2005
%F G.f.: A(x, y) = (1/x)*(1 - 1/(1 + Sum_{n>=1} [Product_{k=0..n-1}(1+k*y)]*x^n )). - _Paul D. Hanna_, Aug 16 2005
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 2;
%e 0, 1, 6, 6;
%e 0, 1, 12, 34, 24;
%e 0, 1, 20, 110, 210, 120;
%e 0, 1, 30, 270, 974, 1452, 720; ...
%t m = 10;
%t gf = (1/x)*(1-1/(1+Sum[Product[(1+k*y), {k, 0, n-1}]*x^n, {n, 1, m}]));
%t CoefficientList[#, y]& /@ CoefficientList[gf + O[x]^m, x] // Flatten (* _Jean-François Alcover_, May 11 2019 *)
%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==0,1,if(k==0,0,polcoeff(polcoeff( (1-1/(1+sum(m=1,n+k,prod(j=0,m-1,1+j*y)*x^m)))/x +x*O(x^n),n,x)+y*O(y^k),k,y)))) \\ _Paul D. Hanna_, Aug 16 2005
%Y Cf. A084938, A111184.
%Y Diagonals: A000007, A000012, A002378, A000142.
%Y Row sums: A003319.
%K easy,nonn,tabl
%O 0,6
%A _Philippe Deléham_, Jan 11 2004
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