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A007442 Inverse binomial transform of primes.

%I A007442 M0065 #95 Jun 27 2025 22:10:59
%S A007442 2,1,1,-1,3,-9,23,-53,115,-237,457,-801,1213,-1389,445,3667,-15081,
%T A007442 41335,-95059,195769,-370803,652463,-1063359,1570205,-1961755,1560269,
%U A007442 1401991,-11023119,36000427,-93408425,214275735,-450374071,879254493,-1599245737,2695465017
%N A007442 Inverse binomial transform of primes.
%C A007442 a(n) is the (n-1)-st difference of the first n primes. Although the magnitude of the terms appears to grow exponentially, a plot shows that the sequence a(n)/2^n has quite a bit of structure. See A082594 for an interesting application. - _T. D. Noe_, May 09 2003
%C A007442 Graph this divided by A122803 using plot2! - _Franklin T. Adams-Watters_
%C A007442 From _Robert G. Wilson v_, Jan 28 2020: (Start)
%C A007442 a(n) is odd for all n>1.
%C A007442 As opposed to A331573, there are terms where abs(a(n)) >= abs(a(n+1)). (End)
%D A007442 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007442 Robert G. Wilson v, <a href="/A007442/b007442.txt">Table of n, a(n) for n = 1..3321</a> (first 1000 from Franklin T. Adams-Watters)
%H A007442 Vaclav Kotesovec, <a href="/A007442/a007442.jpg">Plot of |a(n)|^(1/n) for n = 1..10000</a>
%H A007442 T. D. Noe, <a href="http://www.sspectra.com/math/A007442.gif">Plot of A007442</a>
%H A007442 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A007442 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialTransform.html">Binomial Transform</a>
%F A007442 a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) binomial(n-1, k) prime(k+1).
%F A007442 a(n) = A095195(n,n-1). - _Alois P. Heinz_, Sep 25 2013
%F A007442 G.f.: Sum_{k>=1} prime(k)*x^k/(1 + x)^k. - _Ilya Gutkovskiy_, Apr 23 2019
%e A007442 a(4) = 7 - 3*5 + 3*3 - 2 = -1.
%t A007442 Diff[lst_List] := Table[lst[[i + 1]] - lst[[i]], {i, Length[lst] - 1}]; n=1000; dt = Prime[Range[n]]; a = Range[n]; a[[1]] = 2; Do[dt = Diff[dt]; a[[i]] = dt[[1]], {i, 2, n}]; a
%t A007442 u = Table[Prime[Range[k]], {k, 1, 100}];Flatten[Table[Differences[u[[k]], k - 1], {k, 1, 100}]] (* _Clark Kimberling_, May 15 2015 *)
%t A007442 t = Array[Prime, 30]; f[x_] := Rest[x] - Most[x];
%t A007442 Flatten[Last /@ (NestList[f, t[[1 ;; #]], (# - 1)] & /@ Range[1, 29])] (* _Horst H. Manninger_, Mar 22 2021 *)
%o A007442 (PARI) vector(50, n, sum(k=0, n-1,(-1)^(n-k-1)*binomial(n-1, k)*prime(k+1))) \\ _Altug Alkan_, Oct 17 2015
%Y A007442 Cf. A007443, A082594, A095195, A122803, A331573.
%K A007442 sign,easy
%O A007442 1,1
%A A007442 _N. J. A. Sloane_
%E A007442 Incorrect conjecture concerning the sign of even terms removed by _Glen Whitney_, Nov 10 2024