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A sequence related to binomial(n+4, 4).
2

%I #14 Sep 08 2022 08:45:09

%S 1,7,39,193,886,3858,16146,65502,259119,1003833,3820689,14322663,

%T 52986636,193759452,701265924,2514778812,8943620589,31569189723,

%U 110673119691,385569479997,1335567565746,4601780568342,15778086835014

%N A sequence related to binomial(n+4, 4).

%C Binomial transform of A055589 (without leading 0).

%C 2nd binomial transform of binomial(n+4, 4), A000332.

%C 3rd binomial transform of (1,4,6,4,1,0,0,0,...).

%H G. C. Greubel, <a href="/A081898/b081898.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15,-90,270,-405,243).

%F a(n) = 3^n*(n^4 + 42*n^3 + 515*n^2 + 2034*n + 1944)/1944.

%F G.f.: (1 - 2*x)^4/(1 - 3*x)^5.

%F E.g.f.: (24 + 96*x + 72*x^2 + 16*x^3 + x^4)*exp(3*x)/24. - _G. C. Greubel_, Oct 18 2018

%t LinearRecurrence[{15,-90,270,-405,243}, {1,7,39,193,886}, 50] (* _G. C. Greubel_, Oct 18 2018 *)

%o (PARI) x='x+O('x^30); Vec((1-2*x)^4/(1-3*x)^5) \\ _G. C. Greubel_, Oct 18 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^4/(1-3*x)^5)); // _G. C. Greubel_, Oct 18 2018

%Y Cf. A081899.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Mar 30 2003