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A240707 Sum of the middle parts in the partitions of 4n-1 into 3 parts. 2

%I

%S 1,8,31,80,159,282,459,690,993,1378,1841,2404,3077,3852,4755,5796,

%T 6963,8286,9775,11414,13237,15254,17445,19848,22473,25296,28359,31672,

%U 35207,39010,43091,47418,52041,56970,62169,67692,73549,79700,86203,93068,100251

%N Sum of the middle parts in the partitions of 4n-1 into 3 parts.

%C Original definition: Sum of the second largest parts in the partitions of 4n into 4 parts with smallest part = 1 (see the example).

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

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-4,2,-1,2,-1).

%F G.f.: x*(x^2+3*x+1)*(3*x^4+3*x^3+6*x^2+3*x+1) / ((x-1)^4*(x^2+x+1)^2). - _Colin Barker_, Apr 13 2014

%e For a(n) add the parts in the second columns.

%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 + 1 + 1 + 1 9 + 4 + 2 + 1

%e 8 + 2 + 1 + 1 8 + 5 + 2 + 1

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

%e 6 + 4 + 1 + 1 9 + 3 + 3 + 1

%e 5 + 5 + 1 + 1 8 + 4 + 3 + 1

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

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

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

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

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

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

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

%e 1 8 31 80 .. a(n)

%p A240707:=n->add(add(i*floor((signum((floor((4*n-2-j)/2)-j))+2)/2), i=j+1..floor((4*n-2-j)/2)), j=0..2*n); seq(A240707(n), n=1..50);

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

%o (PARI) Vec(x*(x^2+3*x+1)*(3*x^4+3*x^3+6*x^2+3*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ _Colin Barker_, Apr 13 2014

%o (PARI) A240707(n)=sum(a=1,(4*n-1)\3,(4*n-1-a)\2*((4*n-1-a)\2+1)-a*(a-1))\2 \\ The summand is sum(b=a,(4*n-1-a)\2,b). - _M. F. Hasler_, Apr 17 2014

%Y Cf. A238328, A238340, A238702, A238705, A238706, A239056, A239057, A239059.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Apr 10 2014

%E Definition simplified by _M. F. Hasler_, Apr 17 2014

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Last modified March 29 03:09 EDT 2020. Contains 333104 sequences. (Running on oeis4.)