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A sequence related to binomial(n+6, 6).
2

%I #15 Sep 08 2022 08:45:09

%S 1,11,100,820,6290,46006,324556,2225060,14902075,97873625,632200000,

%T 4025225000,25307562500,157349687500,968628125000,5909609375000,

%U 35763408203125,214838427734375,1281885742187500,7601284179687500

%N A sequence related to binomial(n+6, 6).

%C Binomial transform of A081905.

%C 4th binomial transform of binomial(n+6, 6).

%C 5th binomial transform of (1,6,15,20,15,6,1,0,0,0,...).

%H G. C. Greubel, <a href="/A081906/b081906.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (35,-525,4375,-21875,65625,-109375, 78125).

%F a(n) = 5^n*(n^6 + 165*n^5 + 9535*n^4 + 238575*n^3 + 2590024*n^2 + 10661700*n + 11250000)/11250000.

%F G.f.: (1-4*x)^6/(1-5*x)^7.

%F E.g.f.: (720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(5*x) / 720. - _G. C. Greubel_, Oct 17 2018

%t LinearRecurrence[{35, -525, 4375, -21875, 65625, -109375, 78125}, {1, 11, 100, 820, 6290, 46006, 324556}, 50] (* _G. C. Greubel_, Oct 17 2018 *)

%o (PARI) x='x+O(x^30); Vec((1-4*x)^6/(1-5*x)^7) \\ _G. C. Greubel_, Oct 17 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)^6/(1-5*x)^7)); // _G. C. Greubel_, Oct 17 2018

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 31 2003