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A053539 a(n) = n * 8^(n-1).

%I A053539 #82 Aug 12 2025 12:44:19
%S A053539 0,1,16,192,2048,20480,196608,1835008,16777216,150994944,1342177280,
%T A053539 11811160064,103079215104,893353197568,7696581394432,65970697666560,
%U A053539 562949953421312,4785074604081152,40532396646334464,342273571680157696,2882303761517117440,24211351596743786496
%N A053539 a(n) = n * 8^(n-1).
%C A053539 The Szeged index of the hypercube Q_n (see the Ashrafi et al. reference, p. 45, last line). - _Emeric Deutsch_, Aug 06 2014
%C A053539 For n > 3, 2*a(n) is the number of spanning trees in a superprism on 2*n vertices (see Bogdanowicz). - _Stefano Spezia_, May 05 2024
%D A053539 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A053539 G. C. Greubel, <a href="/A053539/b053539.txt">Table of n, a(n) for n = 0..1000</a>
%H A053539 A. R. Ashrafi, B. Manoochehrian, and H. Yousefi-Azari, <a href="http://bims.iranjournals.ir/article_79.html">On Szeged polynomial of a graph</a>, Bull. Iranian Math. Soc., 33, 2007, 37-46. - _Emeric Deutsch_, Aug 06 2014
%H A053539 Zbigniew R. Bogdanowicz, <a href="https://doi.org/10.47443/dml.2024.004">The number of spanning trees in a superprism</a>, Discrete Math. Lett. 13 (2024) 66-73. See Theorem 3.1.
%H A053539 Frank Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>.
%H A053539 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-64).
%F A053539 a(n) = 16*a(n-1) - 64*a(n-2), with a(0)=0, a(1)=1. - _Emeric Deutsch_, Aug 06 2014
%F A053539 From _G. C. Greubel_, May 16 2019: (Start)
%F A053539 G.f.: x/(1-8*x)^2.
%F A053539 E.g.f.: x*exp(8*x). (End)
%F A053539 From _Amiram Eldar_, Oct 28 2020: (Start)
%F A053539 Sum_{n>=1} 1/a(n) = 8*log(8/7).
%F A053539 Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(9/8). (End)
%p A053539 a := proc(n) option remember; if n<2 then n else 16*a(n-1)-64*a(n-2) end if end proc: seq(a(n), n = 0 .. 20); # _Emeric Deutsch_, Aug 06 2014
%t A053539 Table[n 8^(n-1),{n,0,20}] (* or *) LinearRecurrence[{16,-64},{0,1},20] (* _Harvey P. Dale_, Feb 01 2017 *)
%o A053539 (Magma) [n*8^(n-1): n in [0..20]]; // _Vincenzo Librandi_, Feb 09 2011
%o A053539 (PARI) a(n) = n*8^(n-1); \\ _Joerg Arndt_, Aug 07 2014
%o A053539 (Sage) [n*8^(n-1) for n in (0..20)] # _G. C. Greubel_, May 16 2019
%o A053539 (GAP) List([0..20], n-> n*8^(n-1)); # _G. C. Greubel_, May 16 2019
%Y A053539 Binomial transform of A027473.
%Y A053539 Cf. A001787, A053464, A053469, A053540.
%K A053539 easy,nonn
%O A053539 0,3
%A A053539 _Barry E. Williams_, Jan 15 2000
%E A053539 Offset corrected and name edited by _Emeric Deutsch_, Aug 06 2014