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Sum over each antidiagonal of A248017.
6

%I #35 Oct 22 2018 01:41:48

%S 0,0,0,4,69,554,3100,13288,47492,147050,407568,1030912,2419025,

%T 5324684,11099416,22065120,42085344,77378556,137705904,237996060,

%U 400624581,658434694,1058839380,1669118984,2583424948,3931632406,5890783808,8699293304,12674960961

%N Sum over each antidiagonal of A248017.

%H Christopher Hunt Gribble, <a href="/A248027/b248027.txt">Table of n, a(n) for n = 1..10000</a>

%F Empirically, a(n) = (2*n^11 + 22*n^10 + 22*n^9 - 462*n^8 - 1122*n^7 + 7392*n^6 - 3509*n^5 - 25663*n^4 + 48950*n^3 - 22869*n^2 - 65133*n + 41580 - (693*n^5 + 3465*n^4 - 6930*n^3 - 45045*n^2 + 27027*n + 41580)*(-1)^n)/2661120.

%F Empirical g.f.: -x^4*(x^11 + 2*x^10 - 7*x^9 - 10*x^8 - 28*x^7 - 170*x^6 - 484*x^5 - 538*x^4 - 461*x^3 - 176*x^2 - 45*x - 4) / ((x - 1)^12*(x + 1)^6). - _Colin Barker_, Apr 21 2015

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

%e . Antidiagonals of A248017 n a(n)

%e . 0 1 0

%e . 0 0 2 0

%e . 0 0 0 3 0

%e . 0 2 2 0 4 4

%e . 1 14 39 14 1 5 69

%e . 3 66 208 208 66 3 6 554

%e . 12 198 794 1092 794 198 12 7 3100

%e . 28 508 2196 3912 3912 2196 508 28 8 13288

%e .66 1092 5231 10626 13462 10626 5231 1092 66 9 47492

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

%p (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n

%p + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4

%p - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3

%p - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2

%p + 48*k + 48*n + 45

%p + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3

%p + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n

%p - 45)*(-1)^k

%p + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n

%p - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n - 48*k

%p - 45)*(-1)^n

%p + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;

%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. A248017.

%K nonn

%O 1,4

%A _Christopher Hunt Gribble_, Sep 30 2014

%E Terms corrected and extended by _Christopher Hunt Gribble_, Apr 17 2015