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%I #18 Jul 19 2016 11:32:18
%S 1,0,1,0,1,1,0,2,4,1,0,7,19,11,1,0,38,123,107,26,1,0,295,1076,1195,
%T 474,57,1,0,3098,12350,16198,8668,1836,120,1,0,42271,180729,268015,
%U 176091,52831,6549,247,1,0,726734,3290353,5369639,4105015,1564817,287473,22145
%N Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 1, 3, 3, 6, 6, 10, 10, 15, ...) DELTA (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) where DELTA is the operator defined in A084938.
%H Paul D. Hanna, <a href="/A211183/b211183.txt">Rows n = 0..31, flattened.</a>
%F Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A000012(n), A000366(n+1), A110501(n+1), A211194(n), A221972(n) for x = 0, 1, 2, 3, 4 respectively.
%F T(n,n-1) = A000295(n).
%F T(n,1) = A000366(n).
%F G.f.: A(x,y) = Sum_{n>=0} n! * x^n * Product_{k=1..n} (y + (k-1)/2) / (1 + (k*y + k*(k-1)/2)*x). - _Paul D. Hanna_, Feb 03 2013
%e Triangle begins :
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 4, 1;
%e 0, 7, 19, 11, 1;
%e 0, 38, 123, 107, 26, 1;
%e 0, 295, 1076, 1195, 474, 57, 1;
%e 0, 3098, 12350, 16198, 8668, 1836, 120, 1;
%e 0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1;
%e 0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145, 502, 1; ...
%o (PARI) T(n,k)=polcoeff(polcoeff(sum(m=0, n, m!*x^m*prod(k=1, m, (y + (k-1)/2)/(1+(k*y+k*(k-1)/2)*x+x*O(x^n)))), n,x),k,y)
%o for(n=0,12,for(k=0,n,print1(T(n,k),", "));print()) \\ _Paul D. Hanna_, Feb 03 2013
%Y Cf. A000366, A110501, A211194, A221972.
%K nonn,tabl
%O 0,8
%A _Philippe Deléham_, Feb 02 2013
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