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A229061 The (n+1)-th term of the n-th differences of the prime sequence.

%I A229061 #41 Oct 27 2023 19:15:24
%S A229061 2,2,2,4,8,-2,-48,-70,0,56,308,1014,798,-2072,-5126,-2820,434,-1340,
%T A229061 62902,398032,1247046,2834160,5266626,7862442,9510040,13829302,
%U A229061 37650208,111410394,260524940,468110450,626899146,481007522,-490911164,-3217336656,-8570944960
%N A229061 The (n+1)-th term of the n-th differences of the prime sequence.
%C A229061 All terms are even. The only zero seems to be a(8), corresponding to A036269(9).
%H A229061 Alois P. Heinz, <a href="/A229061/b229061.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 0..98 from Jean-François Alcover)
%F A229061 a(n) = A095195(2*n+1,n).
%e A229061 The sequences of differences begin:
%e A229061 2,      3,   5,   7,  11,  13,   17,  19,  23,  29, ...
%e A229061 1,      2,   2,   4,   2,   4,    2,   4,   6,   2, ...
%e A229061 1,      0,   2,  -2,   2,  -2,    2,   2,  -4,   4, ...
%e A229061 -1,     2,  -4,   4,  -4,   4,    0,  -6,   8,  -6, ...
%e A229061 3,     -6,   8,  -8,   8,  -4,   -6,  14, -14,   6, ...
%e A229061 -9,    14, -16,  16, -12,  -2,   20, -28,  20,  -2, ...
%e A229061 23,   -30,  32, -28,  10,  22,  -48,  48, -22,  -6, ...
%e A229061 -53,   62, -60,  38,  12, -70,   96, -70,  16,  16, ...
%e A229061 115, -122,  98, -26, -82, 166, -166,  86,   0, -28, ...
%e A229061 etc.
%e A229061 Main diagonal begins:
%e A229061 2, 2, 2, 4, 8, -2, -48, -70, 0, 56, ... .
%p A229061 T:= proc(n, k) option remember;
%p A229061       `if`(k=0, ithprime(n), T(n+1, k-1)-T(n, k-1))
%p A229061     end:
%p A229061 a:= n-> T(n+1, n):
%p A229061 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 25 2013
%t A229061 max = 100; row[n_] := Differences[Prime /@ Range[max], n]; Table[row[n], {n, 0, max}] // Diagonal
%Y A229061 Cf. A095195, A036263-A036271, A007442.
%K A229061 sign
%O A229061 0,1
%A A229061 _Jean-François Alcover_, Sep 17 2013