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Sum over each antidiagonal of A248011.
6

%I #23 Nov 30 2016 22:12:33

%S 0,0,3,16,67,204,546,1268,2714,5348,9965,17580,29781,48520,76660,

%T 117624,176196,257976,370503,522456,725175,991540,1337974,1782924,

%U 2349438,3063164,3955601,5061524,6423017,8086224,10106280,12543280,15468232,18958128,23103051,28000224,33762411,40510812,48384906,57534052

%N Sum over each antidiagonal of A248011.

%H Christopher Hunt Gribble, <a href="/A248016/b248016.txt">Table of n, a(n) for n = 1..10000</a>

%F Empirically, a(n) = (2*n^7 + 14*n^6 + 14*n^5 + 70*n^4 - 77*n^3 - 399*n^2 + 61*n + 105 - 105*(-1)^n - 35*n^3*(-1)^n - 105*n^2*(-1)^n + 35*n*(-1)^n)/6720.

%F Empirical g.f.: -x^3*(x^2+1)*(x^4-6*x^2-4*x-3) / ((x-1)^8*(x+1)^4). - _Colin Barker_, Apr 06 2015

%e a(1..9) are formed as follows:

%e . Antidiagonals of A248011 n a(n)

%e . 0 1 0

%e . 0 0 2 0

%e . 1 1 1 3 3

%e . 2 6 6 2 4 16

%e . 6 14 27 14 6 5 67

%e . 10 32 60 60 32 10 6 204

%e . 19 55 129 140 129 55 19 7 546

%e . 28 94 218 294 294 218 94 28 8 1268

%e .44 140 363 506 608 506 363 140 44 9 2714

%p b := proc (n::integer, k::integer)::integer;

%p (4*k^3*n^3 - 12*k^2*n^2 + 2*k^3 + 6*k^2*n + 6*k*n^2 + 2*n^3 - 12*k^2 + 11*k*n - 12*n^2 + 4*k + 4*n - 3 - (2*k^3 + 6*k^2*n - 12*k^2 + 3*k*n + 4*k - 3)*(-1)^n - (6*k*n^2 + 2*n^3 + 3*k*n - 12*n^2 + 4*n - 3)*(-1)^k + (3*k*n - 3)*(-1)^k*(-1)^n)/96;

%p end proc;

%p for j to 10000 do

%p a := 0;

%p for k from j by -1 to 1 do

%p n := j-k+1;

%p a := a+b(n, k)

%p end do;

%p printf("%d, ", a)

%p end do;

%Y Cf. A248011.

%K nonn

%O 1,3

%A _Christopher Hunt Gribble_, Sep 29 2014

%E Terms corrected and extended by _Christopher Hunt Gribble_, Apr 02 2015