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A027276
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a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).
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18
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1, 6, 27, 72, 270, 648, 2268, 5184, 17496, 38880, 128304, 279936, 909792, 1959552, 6298560, 13436928, 42830208, 90699264, 287214336, 604661760, 1904684544, 3990767616, 12516498432, 26121388032, 81629337600, 169789022208
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..2n} (k+1) * A026552(n, k).
G.f.: (1 +6*x +15*x^2 -18*x^3)/(1-6*x^2)^2.
a(n) = -(1/2)*[n=0] + (1/4)*6^(n/2)*(n + 1)*(3*(1 + (-1)^n) + sqrt(6)*(1 - (-1)^n)). - G. C. Greubel, Dec 18 2021
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MATHEMATICA
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Table[-(1/2)*Boole[n==0] + (1/4)*6^(n/2)*(n+1)*(3*(1+(-1)^n) + Sqrt[6]*(1-(-1)^n)), {n, 0, 40}] (* G. C. Greubel, Dec 18 2021 *)
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PROG
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(Magma) I:= [6, 27, 72, 270]; [1] cat [n le 4 select I[n] else 12*(Self(n-2) - 3*Self(n-4)): n in [1..41]]; // G. C. Greubel, Dec 18 2021
(Sage)
@CachedFunction
if (k==0 or k==2*n): return 1
elif (k==1 or k==2*n-1): return (n+2)//2
elif (n%2==0): return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
else: return T(n-1, k) + T(n-1, k-2)
@CachedFunction
def a(n): return sum( (k+1)*T(n, k) for k in (0..2*n) )
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -36, 0, 12, 0]^n*[1; 6; 27; 72])[1, 1] \\ Charles R Greathouse IV, Oct 21 2022
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CROSSREFS
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Cf. A026552, A026553, A026554, A026555, A026556, A026557, A026558, A026559, A026560, A026563, A026564, A026566, A026567, A027272, A027273, A027274, A027275.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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