A240707 - OEIS

%I #25 Jan 22 2018 02:58:49

%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