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A000967 Sum of Fermat coefficients.

%I A000967 M1148 N0437 #33 Sep 08 2022 08:44:28
%S A000967 1,2,4,8,18,40,91,210,492,1165,2786,6710,16267,39650,97108,238824,
%T A000967 589521,1459960,3626213,9030450,22542396,56393792,141358274,354975429,
%U A000967 892893120,2249412290,5674891000,14335757256,36259245522,91815545800
%N A000967 Sum of Fermat coefficients.
%D A000967 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000967 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000967 Reinhard Zumkeller, <a href="/A000967/b000967.txt">Table of n, a(n) for n = 1..1000</a> (corrected by Sean A. Irvine, April 17, 2019)
%H A000967 R. P. Loh, A. G. Shannon, A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, Preprint, 1980.
%H A000967 P. A. Piza, <a href="http://www.jstor.org/stable/3029103">Fermat coefficients</a>, Math. Mag., 27 (1954), 141-146.
%F A000967 Following Piza's definition for the Fermat coefficients: (n:c)=binomial(2n-c, c-1)/c, a(n)= Round( sum_ {c=1..n} (n:c) ). - Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005
%F A000967 a(n) = rounded(sum(A258993(n,k)/(2*k+1)): k = 0..n-1). - _Reinhard Zumkeller_, Jun 22 2015
%e A000967 n...Sum_{c=1..n} (n:c).....a(n)
%e A000967 --------------------------------
%e A000967 .1........1.................1
%e A000967 .2........2.................2
%e A000967 .3........4.................4
%e A000967 .4........8+1/3.............8
%p A000967 FermatCoeff:=(n,c)->binomial(2*n-c,c-1)/c:seq(round(add(FermatCoeff(n,c),c=1..n)),n=1..40); # Pab Ter, Oct 13 2005
%t A000967 Table[Round[Sum[Binomial[n+k, n-k]/(2*k+1), {k, 0, n-1}]], {n,1,35}] (* _G. C. Greubel_, Apr 16 2019 *)
%o A000967 (Haskell)
%o A000967 import Data.Function (on)
%o A000967 a000967 n = round $ sum $
%o A000967             zipWith ((/) `on` fromIntegral) (a258993_row n) [1, 3 ..]
%o A000967 -- _Reinhard Zumkeller_, Jun 22 2015
%o A000967 (PARI) {a(n) = round(sum(k=0,n-1, binomial(n+k,n-k)/(2*k+1)))}; \\ _G. C. Greubel_, Apr 16 2019
%o A000967 (Magma) [Round((&+[Binomial(n+k,n-k)/(2*k+1): k in [0..n-1]])): n in [1..35]]; // _G. C. Greubel_, Apr 16 2019
%o A000967 (Sage) [round(sum(binomial(n+k,n-k)/(2*k+1) for k in (0..n-1))) for n in (1..35)] # _G. C. Greubel_, Apr 16 2019
%Y A000967 Cf. A143858, A258708.
%K A000967 nonn
%O A000967 1,2
%A A000967 _N. J. A. Sloane_
%E A000967 More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 13 2005