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Triangle T(n, k) read by rows; given by [1, 2, 3, 4, 5, 6, ..] DELTA [1, 4, 9, 16, 25, 36, ...] where DELTA is the operator defined in A084938.
7

%I #15 Aug 08 2015 10:43:00

%S 1,1,1,3,8,5,15,75,121,61,105,840,2478,3128,1385,945,11025,51030,

%T 115350,124921,50521,10395,166320,1105335,3859680,7365633,7158128,

%U 2702765

%N Triangle T(n, k) read by rows; given by [1, 2, 3, 4, 5, 6, ..] DELTA [1, 4, 9, 16, 25, 36, ...] where DELTA is the operator defined in A084938.

%H Alois P. Heinz, <a href="/A086872/b086872.txt">Rows n = 0..140, flattened</a>

%F Sum( k>=0, T(n, k)*(-1)^k ) = 0; if n>0.

%F Sum( k>=0, T(n, k)*(-1/2)^k ) = (1/2)^n.

%F Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = (-1)^n*A121822(n), (-1)^n*A092812(n), (-1)^n*A054879(n), A009117(n), A033999(n), A000007(n), A000364(n), A000182(n+1) for x = -6, -5, -4, -3, -2, -1, 0, 1 respectively .

%e Triangle begins:

%e 1;

%e 1, 1;

%e 3, 8, 5;

%e 15, 75, 121, 61;

%e 105, 840, 2478, 3128, 1385;

%e 945, 11025, 51030, 115350, 124921, 50521;

%e 10395, 166320, 1105335, 3859680, 7365633, 7158128, 2702765 ; ...

%Y Cf. A000182 (row sums), A000364 (first diagonal), A001147 (first column), A084938, A261065 (2nd column).

%K easy,nonn,tabl

%O 0,4

%A _Philippe Deléham_, Aug 20 2003, Aug 17 2007