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%I #8 Aug 10 2018 02:41:11
%S 1,1,1,1,3,1,1,4,4,1,1,6,7,6,1,1,7,14,14,7,1,1,9,20,33,20,9,1,1,10,31,
%T 56,56,31,10,1,1,12,40,97,111,97,40,12,1,1,13,55,142,217,217,142,55,
%U 13,1,1,15,67,213,358,463,358,213,67,15,1,1,16,86,287,590,841,841,590,287,86,16,1
%N 2*A007318 - (A046854 + A065941 - A000012).
%C Row sums = A131403: (1, 2, 5, 10, 21, 44, 93, ...).
%H Andrew Howroyd, <a href="/A131402/b131402.txt">Table of n, a(n) for n = 0..1274</a>
%F 2*A007318 - (A046854 + A065941 - A000012) as infinite lower triangular matrices.
%F T(n,k) = 2*binomial(n, k) + 1 - binomial(floor(n + k)/2), k) - binomial(n-floor((k+1)/2), floor(k/2)). - _Andrew Howroyd_, Aug 09 2018
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 4, 4, 1;
%e 1, 6, 7, 6, 1;
%e 1, 7, 14, 14, 7, 1;
%e 1, 9, 20, 33, 20, 9, 1;
%e 1, 10, 31, 56, 56, 31, 10, 1;
%e ...
%o (PARI) T(n,k) = if(k <= n, 2*binomial(n, k) + 1 - binomial((n + k)\2, k) - binomial(n-(k+1)\2, k\2), 0) \\ _Andrew Howroyd_, Aug 09 2018
%Y Row sums are A131403.
%Y Cf. A046854, A065941.
%K nonn,tabl
%O 0,5
%A _Gary W. Adamson_, Jul 07 2007
%E Missing terms inserted and a(55) and beyond from _Andrew Howroyd_, Aug 09 2018
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