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%I #27 Sep 08 2022 08:45:07
%S 1,10,63,324,1485,6318,25515,99144,373977,1377810,4979799,17714700,
%T 62178597,215765046,741360195,2525407632,8537599665,28669116186,
%U 95692860783,317684800980,1049522104701,3451916556990,11307641812443
%N Binomial transform of n^2*2^n/2.
%C With a leading zero, this is second binomial transform of the hexagonal numbers A000384 (with leading zero). - _Paul Barry_, Jun 09 2003
%C Coefficients in the hypergeometric series identity 1 - 10*x/(x + 9) + 63*x*(x - 1)/((x + 9)*(x + 12)) - 324*x*(x - 1)*(x - 2)/((x + 9)*(x + 12)*(x + 15)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A276289 and A084901. - _Peter Bala_, May 30 2019
%H G. C. Greubel, <a href="/A077616/b077616.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 (9,-27,27).
%F E.g.f: x*(1+2*x)*exp(3*x).
%F O.g.f: ((1/3)*x^(3/4)*3^(3/4)/(-(3*x+1)/(3*x-1)+1)^(1/4))*(-(3*x+1)/(3*x-1)-1)^(1/4)*hypergeom([ -1, 2], [3/2], 3*x/(3*x-1))/(3*x-1)^2, which can also be represented as associated Legendre function: 1/6*x^(3/4)*Pi^(1/2)*3^(3/4)*LegendreP(1, -1/2, (3*x+1)/(1-3*x))/(3*x-1)^2.
%F G.f.: x*(1+x)/(1-3*x)^3. - _Paul Barry_, Jun 09 2003
%F a(n) = n*(2*n+1)*3^(n-2). - _Paul Barry_, Jul 24 2003
%t LinearRecurrence[{9, -27, 27}, {1, 10, 63}, 30] (* _Jean-François Alcover_, May 23 2016 *)
%o (PARI) a(n)=n*(2*n+1)*3^(n-2) \\ _Charles R Greathouse IV_, Mar 19 2017
%o (Magma) [3^(n-2)*n*(1+2*n): n in [1..30]]; // _G. C. Greubel_, Jun 03 2019
%o (Sage) [3^(n-2)*n*(1+2*n) for n in (1..30)] # _G. C. Greubel_, Jun 03 2019
%o (GAP) List([1..30], n-> 3^(n-2)*n*(1+2*n)) # _G. C. Greubel_, Jun 03 2019
%Y Cf. A084901, A276289.
%K nonn,easy
%O 1,2
%A _Karol A. Penson_, Nov 12 2002
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