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%I #58 Oct 27 2023 22:00:45
%S 1,5,16,44,112,272,640,1472,3328,7424,16384,35840,77824,167936,360448,
%T 770048,1638400,3473408,7340032,15466496,32505856,68157440,142606336,
%U 297795584,620756992,1291845632,2684354560,5570035712,11542724608,23890755584,49392123904
%N a(n) = (3*n-1) * 2^(n-2).
%C Coefficients in the hypergeometric series identity 1 - 5*x/(x + 4) + 16*x*(x - 1)/((x + 4)*(x + 6)) - 44*x*(x - 1)*(x - 2)/((x + 4)*(x + 6)*(x + 8)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289. - _Peter Bala_, May 30 2019
%H Vincenzo Librandi, <a href="/A053220/b053220.txt">Table of n, a(n) for n = 1..500</a>
%H Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, <a href="https://arxiv.org/abs/2303.09898">Hypercubes and Isometric Words based on Swap and Mismatch Distance</a>, arXiv:2303.09898 [math.CO], 2023.
%H F. K. Hwang and C. L. Mallows, <a href="http://dx.doi.org/10.1016/0097-3165(95)90097-7">Enumerating nested and consecutive partitions</a>, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F G.f.: x*(1+x)/(1-2*x)^2.
%F a(n) = (3*n-1) * 2^(n-2).
%F E.g.f.: exp(2*x)*(1+3*x). The sequence 0, 1, 5, 16, ... has a(n) = ((3n-1)*2^n + 0^n)/4 (offset 0). It is the binomial transform of A032766. The sequence 1, 5, 16, ... has a(n) = (2+3n)*2^(n-1) (offset 0). It is the binomial transform of A016777. - _Paul Barry_, Jul 23 2003
%F Row sums of A132776(n-1). - _Gary W. Adamson_, Aug 29 2007
%F a(n+1) = det(f(i-j+1))_{1 <= i, j <= n}, where f(0) = 1, f(1) = 5 and for k > 0, we have f(k+1) = 9 and f(-k) = 0. - _Mircea Merca_, Jun 23 2012
%t ListCorrelate[{1, 1}, Table[n 2^(n - 1), {n, 0, 28}]] (* or *) ListConvolve[{1, 1}, Table[n 2^(n - 1), {n, 0, 28}]] (* _Ross La Haye_, Feb 24 2007 *)
%t LinearRecurrence[{4, -4}, {1, 5}, 35] (* _Vladimir Joseph Stephan Orlovsky_, Jan 29 2012 *)
%t Array[(3# - 1) 2^(# - 2) &, 35] (* _Alonso del Arte_, Sep 04 2018 *)
%t CoefficientList[Series[(1 + x)/(1 - 2 * x)^2, {x, 0, 50}], x] (* _Stefano Spezia_, Sep 04 2018 *)
%o (PARI) a(n)=if(n<1,0,(3*n-1)*2^(n-2))
%o (PARI) a(n)=(3*n-1)<<(n-2) \\ _Charles R Greathouse IV_, Apr 17 2012
%o (Magma) [(3*n-1)*2^(n-2): n in [1..50]]; // _Vincenzo Librandi_, May 09 2011
%o (Haskell)
%o a053220 n = a056242 (n + 1) n -- _Reinhard Zumkeller_, May 08 2014
%Y Cf. A053219, A053221, A132776, A276289.
%Y Center elements from triangle A053218. Also a diagonal of triangle A056242.
%K nonn,easy
%O 1,2
%A _Asher Auel_, Jan 01 2000
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