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A379781 a(1)=1, a(2)=2; thereafter, a(n) is the final value at the bottom of the difference triangle of the sequence thus far.

%I A379781 #19 Jan 07 2025 10:10:43
%S A379781 1,2,1,-2,0,7,-14,-12,155,-408,-364,7693,-30940,10712,637701,-4224222,
%T A379781 9980180,61922567,-810337234,4100137008,-958593005,-174952472228,
%U A379781 1662063951016,-6944673371867,-22887336602200,655644589917172,-5694691183524699,19946666531550638,176993602416669640
%N A379781 a(1)=1, a(2)=2; thereafter, a(n) is the final value at the bottom of the difference triangle of the sequence thus far.
%C A379781 The difference triangle refers to the triangular array of iterated differences.
%C A379781 The first term in each row of the difference triangle is the inverse binomial transform of the sequence, so the definition means the inverse binomial transform deletes term a(2) = 2.
%H A379781 Neal Gersh Tolunsky, <a href="/A379781/b379781.txt">Table of n, a(n) for n = 1..595</a>
%e A379781 To find a(6), we look at the first difference triangle of the first 5 terms:
%e A379781   1,   2,   1,  -2,   0
%e A379781   1,  -1,  -3,   2
%e A379781  -2,  -2,   5
%e A379781   0,   7
%e A379781   7
%e A379781 7 is the final value, so a(6)=7.
%t A379781 a[1] = 1; a[2] = 2; a[n_] := a[n] = Sum[(-1)^(n-k+1) * Binomial[n-2, k-1] * a[k], {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* _Amiram Eldar_, Jan 04 2025 *)
%Y A379781 Cf. A014182, A010739.
%K A379781 sign
%O A379781 1,2
%A A379781 _Neal Gersh Tolunsky_, Jan 02 2025