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Expansion of 1/((1+x)*(1-x)^7).
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%I #45 Jan 18 2024 09:51:19

%S 1,6,22,62,148,314,610,1106,1897,3108,4900,7476,11088,16044,22716,

%T 31548,43065,57882,76714,100386,129844,166166,210574,264446,329329,

%U 406952,499240,608328,736576,886584,1061208,1263576,1497105,1765518,2072862,2423526,2822260,3274194,3784858

%N Expansion of 1/((1+x)*(1-x)^7).

%H Vincenzo Librandi, <a href="/A001769/b001769.txt">Table of n, a(n) for n = 0..10000</a>

%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,14,0,-14,14,-6,1).

%F From _Paul Barry_, Jul 01 2003: (Start)

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+6, 6).

%F a(n) = (4*n^6 +96*n^5 +910*n^4 +4320*n^3 +10696*n^2 +12864*n+5715)/5760+(-1)^n/128. (End)

%F Boas-Buck recurrence: a(n) = (1/n)*Sum_{p=0..n-1} (7 + (-1)^(n-p))*a(p), n >= 1, a(0) = 1. See the Boas-Buck comment in A046521 (here for the unsigned column k = 3 with offset 0).

%F a(n)+a(n+1) = A000579(n+7). - _R. J. Mathar_, Jan 06 2021

%t CoefficientList[Series[1/((1+x)(1-x)^7),{x,0,30}],x] (* or *) LinearRecurrence[ {6,-14,14,0,-14,14,-6,1},{1,6,22,62,148,314,610,1106},40] (* _Harvey P. Dale_, May 24 2015 *)

%o (Magma) [(4*n^6+96*n^5+910*n^4+4320*n^3+10696*n^2+12864*n+5715)/5760+(-1)^n/128: n in [0..40]]; // _Vincenzo Librandi_, Aug 15 2011

%o (PARI) a(n)=(4*n^6+96*n^5+910*n^4+4320*n^3+10696*n^2+12864*n)\/5760+1 \\ _Charles R Greathouse IV_, Apr 17 2012

%Y Cf. A002620, A002623, A001752, A001753 (first differences), A158454 (signed column k=3), A001779 (partial sums), A169794 (binomial transf.).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_