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%I #12 Sep 08 2022 08:45:33
%S 1,-4,3,3,-8,5,5,-12,7,7,-16,9,9,-20,11,11,-24,13,13,-28,15,15,-32,17,
%T 17,-36,19,19,-40,21,21,-44,23,23,-48,25,25,-52,27,27,-56,29,29,-60,
%U 31,31,-64,33,33,-68,35,35,-72,37,37,-76,39,39,-80,41
%N Hankel transform of binomial(2*n+3, n).
%C Hankel transform of A002054(n+1).
%C Hankel transform of A002054(n) is A057078(n+1).
%C Partial sums are A138188.
%H G. C. Greubel, <a href="/A138187/b138187.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-3,-2,-1).
%F G.f.: (1 -2*x -2*x^2 -x^3)/(1 +x +x^2)^2.
%F a(n) = Sum_{k=0..n} (-1)^(n-k+1)*(n+k+2)*binomial(n+k+1, 2*k). - _Paul Barry_, Apr 19 2010
%F a(n) = 2*floor(n/3) + 1 if (n mod 3) = 0, -4*(floor(n/3) + 1) if (n mod 3) = 1 and 2*floor(n/3) + 3 if (n mod 3) = 2. - _G. C. Greubel_, Jun 16 2021
%t a[n_]:= a[n]= Sum[(-1)^(n-k+1)*(n+k+2)*Binomial[n+k+1, 2*k], {k, 0, n+1}];
%t Table[a[n], {n, 0, 65}] (* _G. C. Greubel_, Jun 16 2021 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1-2*x-2*x^2-x^3)/(1+x+x^2)^2 )); // _G. C. Greubel_, Jun 16 2021
%o (Sage)
%o @CachedFunction
%o def A138187(n):
%o if (n%3==0): return 2*(n//3) +1
%o elif (n%3==1): return -4*((n//3) +1)
%o else: return 2*(n//3) +3
%o [A138187(n) for n in (0..65)] # _G. C. Greubel_, Jun 16 2021
%Y Cf. A002054, A057078, A138188.
%K easy,sign
%O 0,2
%A _Paul Barry_, Mar 04 2008
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