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%I #16 Sep 08 2022 08:44:49
%S 1,4,10,28,64,172,388,1036,2332,6220,13996,37324,83980,223948,503884,
%T 1343692,3023308,8062156,18139852,48372940,108839116,290237644,
%U 653034700,1741425868,3918208204,10448555212,23509249228,62691331276
%N a(n) = Sum_{i=0..2*n} Sum_{j=0..n-1} A026519(j, i).
%H G. C. Greubel, <a href="/A026534/b026534.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6).
%F a(n) = Sum_{i=0..2*n} Sum_{j=0..n-1} A026519(j, i).
%F G.f.: x*(1+3*x)/((1-x)*(1-6*x^2)). - _Ralf Stephan_, Feb 03 2004
%F a(n) = (1/60)*( 6^((n+1)/2)*( (4*sqrt(6) - 9)*(-1)^n + (4*sqrt(6) + 9) ) - 48 ). - _G. C. Greubel_, Dec 20 2021
%t LinearRecurrence[{1,6,-6}, {1,4,10}, 40] (* _G. C. Greubel_, Dec 20 2021 *)
%o (Magma) I:=[1,4,10]; [n le 3 select I[n] else Self(n-1) +6*Self(n-2) -6*Self(n-3): n in [1..40]]; // _G. C. Greubel_, Dec 20 2021
%o (Sage)
%o @CachedFunction
%o def T(n, k): # T = A026519
%o if (k<0 or k>2*n): return 0
%o elif (k==0 or k==2*n): return 1
%o elif (k==1 or k==2*n-1): return (n+1)//2
%o elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
%o else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
%o @CachedFunction
%o def a(n): return sum( sum( T(j,i) for i in (0..2*n) ) for j in (0..n-1) )
%o [a(n) for n in (1..40)]
%o (PARI) Vec((1+3*x)/((1-x)*(1-6*x^2))+O(x^99)) \\ _Charles R Greathouse IV_, Jan 24 2022
%Y Cf. A026519, A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A027262, A027263, A027264, A027265, A027266.
%Y Cf. A026532, A026551.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_
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