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A081905 Third binomial transform of binomial(n+6, 6).

%I A081905 #19 Mar 22 2025 15:25:01
%S A081905 1,10,79,552,3567,21810,127905,725820,4009920,21664000,114840064,
%T A081905 598865920,3078537216,15626600448,78431059968,389685706752,
%U A081905 1918516592640,9367021682688,45387134009344,218388081147904,1044061452500992,4961718019031040,23449374679891968,110252343064264704
%N A081905 Third binomial transform of binomial(n+6, 6).
%C A081905 Binomial transform of A081904.
%C A081905 4th binomial transform of (1,6,15,20,15,6,1,0,0,0,...).
%H A081905 G. C. Greubel, <a href="/A081905/b081905.txt">Table of n, a(n) for n = 0..1000</a>
%H A081905 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (28,-336,2240,-8960,21504,-28672,16384).
%F A081905 a(n) = 4^n*(n^6 + 129*n^5 + 5845*n^4 + 115215*n^3 + 993874*n^2 + 3308616*n + 2949120)/2949120.
%F A081905 G.f.: (1-3*x)^6/(1-4*x)^7.
%F A081905 a(n) = 28*a(n-1) - 336*a(n-2) + 2240*a(n-3) - 8960*a(n-4) + 21504*a(n-5) - 28672*a(n-6) + 16384*a(n-7); a(0)=1, a(1)=10, a(2)=79, a(3)=552, a(4)=3567, a(5)=21810, a(6)=127905. - _Harvey P. Dale_, Aug 14 2014
%F A081905 E.g.f.: (720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(4*x) / 720. - _G. C. Greubel_, Oct 17 2018
%t A081905 CoefficientList[Series[(1-3x)^6/(1-4x)^7,{x,0,20}],x] (* or *) LinearRecurrence[{28,-336,2240,-8960,21504,-28672,16384},{1,10,79,552,3567,21810,127905},20] (* _Harvey P. Dale_, Aug 14 2014 *)
%o A081905 (PARI) x='x+O(x^30); Vec((1-3*x)^6/(1-4*x)^7) \\ _G. C. Greubel_, Oct 17 2018
%o A081905 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)^6/(1-4*x)^7)); // _G. C. Greubel_, Oct 17 2018
%Y A081905 Cf. A000579, A081904, A081906.
%K A081905 easy,nonn
%O A081905 0,2
%A A081905 _Paul Barry_, Mar 31 2003