A211183 - OEIS

%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