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A130002 Alternating sum along antidiagonals of the array of A129952 and its iterated differences.

%I A130002 #9 Jan 30 2018 21:13:11
%S A130002 1,1,2,3,10,23,60,139,326,735,1648,3635,7962,17287,37316,80091,171118,
%T A130002 364079,771864,1631107,3436994,7223511,15146092,31690283,66176790,
%U A130002 137945983,287076800,596523219,1237785706,2565049895,5309056788,10976027515,22667882942
%N A130002 Alternating sum along antidiagonals of the array of A129952 and its iterated differences.
%C A130002 Define the square array T of A129952 and its iterated differences: T(0,n)=A129952(n), T(d,n)=T(d-1,n+1)-T(d-1,n), d>0. Then a(n) = sum_{d=0..n} (-1)^d*T(d,n-d), the sum along the antidiagonals of T(d,n), alternating signs.
%F A130002 From _Chai Wah Wu_, Jan 30 2018: (Start)
%F A130002 a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 7*a(n-4) + 4*a(n-5) - 4*a(n-6) for n > 6.
%F A130002 G.f.: (-2*x^6 - 6*x^5 + 3*x^4 + 5*x^3 - 3*x + 1)/((x - 1)^2*(x + 1)^2*(2*x - 1)^2). (End)
%e A130002 The original series and first, 2nd etc. differences are the rows of
%e A130002 1..1..2...6..16..40.. <- A129952 = T(0,n)
%e A130002 0..1..4..10..24..56.. <- A129953 = T(1,n)
%e A130002 1..3..6..14..32..72.. <- A129954 = T(2,n)
%e A130002 2..3..8..18..40..88.. <- A129955 = T(3,n)
%e A130002 1..5.10..22..48......
%e A130002 ...
%e A130002 a(2) = 2-1+1 = 2. a(3) = 6-4+3-2 = 3. a(4) = 16-10+6-3+1 = 10.
%K A130002 nonn
%O A130002 0,3
%A A130002 _Paul Curtz_, Jun 15 2007
%E A130002 Edited and extended by _R. J. Mathar_, Jun 30 2008