Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #77 Aug 13 2024 05:04:32
%S 0,1,16,192,2048,20480,196608,1835008,16777216,150994944,1342177280,
%T 11811160064,103079215104,893353197568,7696581394432,65970697666560,
%U 562949953421312,4785074604081152,40532396646334464,342273571680157696,2882303761517117440,24211351596743786496
%N a(n) = n * 8^(n-1).
%C The Szeged index of the hypercube Q_n (see the Ashrafi et al. reference, p. 45, last line). - _Emeric Deutsch_, Aug 06 2014
%C 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 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H G. C. Greubel, <a href="/A053539/b053539.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 Zbigniew R. Bogdanowicz, <a href="https://www.dmlett.com/archive/v13/DML24_v13_pp66-73.pdf">The number of spanning trees in a superprism</a>, Discrete Math. Lett. 13 (2024) 66-73. See Theorem 3.1.
%H Frank Ellermann, <a href="/A001792/a001792.txt">Illustration of binomial transforms</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-64).
%F a(n) = 16*a(n-1) - 64*a(n-2), with a(0)=0, a(1)=1. - _Emeric Deutsch_, Aug 06 2014
%F From _G. C. Greubel_, May 16 2019: (Start)
%F G.f.: x/(1-8*x)^2.
%F E.g.f.: x*exp(8*x). (End)
%F From _Amiram Eldar_, Oct 28 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 8*log(8/7).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(9/8). (End)
%p 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 Table[n 8^(n-1),{n,0,20}] (* or *) LinearRecurrence[{16,-64},{0,1},20] (* _Harvey P. Dale_, Feb 01 2017 *)
%o (Magma) [n*8^(n-1): n in [0..20]]; // _Vincenzo Librandi_, Feb 09 2011
%o (PARI) a(n) = n*8^(n-1); \\ _Joerg Arndt_, Aug 07 2014
%o (Sage) [n*8^(n-1) for n in (0..20)] # _G. C. Greubel_, May 16 2019
%o (GAP) List([0..20], n-> n*8^(n-1)) # _G. C. Greubel_, May 16 2019
%Y Binomial transform of A027473.
%Y Cf. A001787, A053464, A053469, A053540.
%K easy,nonn
%O 0,3
%A _Barry E. Williams_, Jan 15 2000
%E Offset corrected and name edited by _Emeric Deutsch_, Aug 06 2014