A229061 - OEIS

%I #41 Oct 27 2023 19:15:24

%S 2,2,2,4,8,-2,-48,-70,0,56,308,1014,798,-2072,-5126,-2820,434,-1340,

%T 62902,398032,1247046,2834160,5266626,7862442,9510040,13829302,

%U 37650208,111410394,260524940,468110450,626899146,481007522,-490911164,-3217336656,-8570944960

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

%C All terms are even. The only zero seems to be a(8), corresponding to A036269(9).

%H 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 a(n) = A095195(2*n+1,n).

%e The sequences of differences begin:

%e 2,      3,   5,   7,  11,  13,   17,  19,  23,  29, ...

%e 1,      2,   2,   4,   2,   4,    2,   4,   6,   2, ...

%e 1,      0,   2,  -2,   2,  -2,    2,   2,  -4,   4, ...

%e -1,     2,  -4,   4,  -4,   4,    0,  -6,   8,  -6, ...

%e 3,     -6,   8,  -8,   8,  -4,   -6,  14, -14,   6, ...

%e -9,    14, -16,  16, -12,  -2,   20, -28,  20,  -2, ...

%e 23,   -30,  32, -28,  10,  22,  -48,  48, -22,  -6, ...

%e -53,   62, -60,  38,  12, -70,   96, -70,  16,  16, ...

%e 115, -122,  98, -26, -82, 166, -166,  86,   0, -28, ...

%e etc.

%e Main diagonal begins:

%e 2, 2, 2, 4, 8, -2, -48, -70, 0, 56, ... .

%p T:= proc(n, k) option remember;

%p       `if`(k=0, ithprime(n), T(n+1, k-1)-T(n, k-1))

%p     end:

%p a:= n-> T(n+1, n):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 25 2013

%t max = 100; row[n_] := Differences[Prime /@ Range[max], n]; Table[row[n], {n, 0, max}] // Diagonal

%Y Cf. A095195, A036263-A036271, A007442.

%K sign

%O 0,1

%A _Jean-François Alcover_, Sep 17 2013