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%I #32 Sep 08 2022 08:45:11
%S 0,1,15,150,1250,9375,65625,437500,2812500,17578125,107421875,
%T 644531250,3808593750,22216796875,128173828125,732421875000,
%U 4150390625000,23345947265625,130462646484375,724792480468750,4005432128906250
%N a(n) = 5^(n-1)*n*(n+1)/2.
%C Binomial transform of A084901. Fourth binomial transform of heptagonal numbers A000566. Fifth binomial transform of (0,1,5,0,0,0,...).
%C Number of n-permutations of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly two u's. Example: a(2)=15 because we have uuw, uuv, uuz, uux, uuy, uwu, uvu, uzu, uxu, uyu, wuu, vuu, zuu, xuu, yuu. - _Zerinvary Lajos_, Dec 30 2007
%C A shifted version of A081135. - _R. J. Mathar_, Sep 17 2008
%H G. C. Greubel, <a href="/A084902/b084902.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-75,125).
%F G.f.: x/(1 - 5*x)^3.
%F E.g.f.: (x/2)*(2 + 5*x)*exp(5*x). - _G. C. Greubel_, May 17 2021
%F a(n) = 15*a(n-1) - 75*a(n-2) + 125*a(n-3). - _Wesley Ivan Hurt_, May 17 2021
%t Table[5^(n-1)n(n+1)/2,{n,0,30}] (* or *) LinearRecurrence[{15,-75,125},{0,1,15},30] (* _Harvey P. Dale_, Sep 18 2018 *)
%o (PARI) a(n)=5^(n-1)*n*(n+1)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [5^(n-1)*Binomial(n+1,2): n in [0..30]]; // _G. C. Greubel_, May 17 2021
%o (Sage) [5^(n-1)*binomial(n+1,2) for n in (0..30)] # _G. C. Greubel_, May 17 2021
%Y Cf. A000566, A038243, A081135, A084901.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Jun 10 2003
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