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Number of partitions of 4n into 4 parts with smallest part = 1.
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%I #36 Dec 29 2021 15:21:54

%S 1,4,10,19,30,44,61,80,102,127,154,184,217,252,290,331,374,420,469,

%T 520,574,631,690,752,817,884,954,1027,1102,1180,1261,1344,1430,1519,

%U 1610,1704,1801,1900,2002,2107,2214,2324,2437,2552,2670,2791,2914,3040,3169

%N Number of partitions of 4n into 4 parts with smallest part = 1.

%C The number of partitions of 4*(n-1) into at most 3 parts. - _Colin Barker_, Mar 31 2015

%H Vincenzo Librandi, <a href="/A238705/b238705.txt">Table of n, a(n) for n = 1..200</a>

%H A. Osorio, <a href="http://mpra.ub.uni-muenchen.de/56690/1/MPRA_paper_56690.pdf">A Sequential Allocation Problem: The Asymptotic Distribution of Resources</a>, Munich Personal RePEc Archive, 2014.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%F G.f.: -x*(x+1)*(2*x^2+x+1) / ((x-1)^3*(x^2+x+1)). - _Colin Barker_, Mar 10 2014

%F a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5). - _Wesley Ivan Hurt_, Nov 18 2021

%e Count the 1's in the last column for a(n):

%e 13 + 1 + 1 + 1

%e 12 + 2 + 1 + 1

%e 11 + 3 + 1 + 1

%e 10 + 4 + 1 + 1

%e 9 + 5 + 1 + 1

%e 8 + 6 + 1 + 1

%e 7 + 7 + 1 + 1

%e 11 + 2 + 2 + 1

%e 10 + 3 + 2 + 1

%e 9 + 4 + 2 + 1

%e 8 + 5 + 2 + 1

%e 7 + 6 + 2 + 1

%e 9 + 3 + 3 + 1

%e 8 + 4 + 3 + 1

%e 7 + 5 + 3 + 1

%e 6 + 6 + 3 + 1

%e 7 + 4 + 4 + 1

%e 6 + 5 + 4 + 1

%e 5 + 5 + 5 + 1

%e 9 + 1 + 1 + 1 10 + 2 + 2 + 2

%e 8 + 2 + 1 + 1 9 + 3 + 2 + 2

%e 7 + 3 + 1 + 1 8 + 4 + 2 + 2

%e 6 + 4 + 1 + 1 7 + 5 + 2 + 2

%e 5 + 5 + 1 + 1 6 + 6 + 2 + 2

%e 7 + 2 + 2 + 1 8 + 3 + 3 + 2

%e 6 + 3 + 2 + 1 7 + 4 + 3 + 2

%e 5 + 4 + 2 + 1 6 + 5 + 3 + 2

%e 5 + 3 + 3 + 1 6 + 4 + 4 + 2

%e 4 + 4 + 3 + 1 5 + 5 + 4 + 2

%e 5 + 1 + 1 + 1 6 + 2 + 2 + 2 7 + 3 + 3 + 3

%e 4 + 2 + 1 + 1 5 + 3 + 2 + 2 6 + 4 + 3 + 3

%e 3 + 3 + 1 + 1 4 + 4 + 2 + 2 5 + 5 + 3 + 3

%e 3 + 2 + 2 + 1 4 + 3 + 3 + 2 5 + 4 + 4 + 3

%e 1 + 1 + 1 + 1 2 + 2 + 2 + 2 3 + 3 + 3 + 3 4 + 4 + 4 + 4

%e 4(1) 4(2) 4(3) 4(4) .. 4n

%e ------------------------------------------------------------------------

%e 1 4 10 19 .. a(n)

%t a[1] = 4; a[n_] := (n/(n - 1))*a[n - 1] + 4 n*Sum[(Floor[(4 n - 2 - i)/2] - i)*(Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; b[n_] := a[n]/(4 n); Table[b[n] - b[n - 1], {n, 50}]

%t LinearRecurrence[{2,-1,1,-2,1},{1,4,10,19,30},50] (* _Harvey P. Dale_, Jun 13 2015 *)

%t Table[Count[IntegerPartitions[4 n,{4}],_?(#[[-1]]==1&)],{n,50}] (* _Harvey P. Dale_, Dec 29 2021 *)

%o (PARI) Vec(-x*(x+1)*(2*x^2+x+1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ _Colin Barker_, Sep 22 2014

%Y Cf. A238328, A238340, A238702.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_ and _Antonio Osorio_, Mar 03 2014