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a(n) = 2^n * (n^4 - 4*n^3 + 18*n^2 - 52*n + 75) - 75.
4

%I #22 Mar 08 2023 18:01:44

%S 0,1,33,357,2405,12405,53877,207541,731829,2411445,7531445,22523829,

%T 64991157,181977013,496680885,1326120885,3473604533,8947236789,

%U 22706651061,56869519285,140755599285,344683708341,835954147253,2009692372917,4792831180725,11346431180725

%N a(n) = 2^n * (n^4 - 4*n^3 + 18*n^2 - 52*n + 75) - 75.

%C This sequence is related to A036828 by the transform a(n) = n*A036828(n) - sum(A036828(i), i=0..n-1).

%H Bruno Berselli, <a href="/A209359/b209359.txt">Table of n, a(n) for n = 0..1000</a>

%H B. Berselli, A description of the transform in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (11,-50,120,-160,112,-32).

%F G.f.: x*(1+2*x)*(1+20*x+4*x^2)/((1-x)*(1-2*x)^5).

%F a(n) = (1/2) * Sum_{k=0..n} Sum_{i=0..n} k^4 * C(k,i). - _Wesley Ivan Hurt_, Sep 21 2017

%t LinearRecurrence[{11, -50, 120, -160, 112, -32}, {0, 1, 33, 357, 2405, 12405}, 26]

%t Table[2^n(n^4-4n^3+18n^2-52n+75)-75,{n,0,30}] (* _Harvey P. Dale_, Mar 08 2023 *)

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1+2*x)*(1+20*x+4*x^2)/((1-x)*(1-2*x)^5)));

%o (PARI) for(n=0, 25, print1(2^n*(n^4-4*n^3+18*n^2-52*n+75)-75", "));

%Y Cf. A000079, A000337, A036826, A036828.

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Mar 07 2012