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A246581 G.f.: x^((k^2 + k)/2) / (Product_{i=1..k} (1 - x^i) * Product_{r>=1} (1 + x^r)) with k = 2.

%I A246581 #35 Oct 13 2017 21:30:09
%S A246581 0,0,0,1,0,1,-1,1,-2,1,-3,2,-4,3,-5,5,-6,7,-8,10,-10,13,-13,17,-17,21,
%T A246581 -22,27,-28,33,-36,41,-45,50,-56,62,-69,75,-85,92,-103,111,-125,135,
%U A246581 -150,162,-180,195,-215,232,-256,278,-303,329,-359,390,-423
%N A246581 G.f.: x^((k^2 + k)/2) / (Product_{i=1..k} (1 - x^i) * Product_{r>=1} (1 + x^r)) with k = 2.
%C A246581 Empirical: 2*(-1)^n*a(n+1) is equal to the number of partitions mu of n such that the diagram of mu and the diagram of the transpose of mu have exactly n-1 cells in common (see below example). - _John M. Campbell_, Feb 01 2016
%H A246581 Vaclav Kotesovec, <a href="/A246581/b246581.txt">Table of n, a(n) for n = 0..5000</a>
%H A246581 J. Fulman, <a href="http://dx.doi.org/10.1090/S0273-0979-01-00920-X">Random matrix theory over finite fields</a>, Bull. Amer. Math. Soc., 39 (No. 1, 2002), 51-85, MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=2.
%F A246581 G.f.: x^3/((1-x)*(1-x^2)) * Product_{k>=1} 1/(1+x^k). - _Vaclav Kotesovec_, Mar 12 2016
%F A246581 a(n) ~ (-1)^(n+1) * 3^(1/4) * exp(sqrt(n/6)*Pi) / (2^(9/4)*Pi*n^(1/4)). - _Vaclav Kotesovec_, Mar 12 2016
%e A246581 From _John M. Campbell_, Feb 01 2016: (Start)
%e A246581 For example, letting n=9, there are 2*(-1)^n*a(n+1) = (-2)*(-3) = 6 partitions mu of n=9 such that the diagram of mu and the diagram of the transpose of mu have exactly n-1 cells in common: (5,2,1,1), (4,3,2), (4,3,1,1), (4,2,2,1), (4,2,1,1,1), (3,3,2,1). For example, the diagram of (3,3,2,1) is
%e A246581    ooo
%e A246581    ooo
%e A246581    oo
%e A246581    o
%e A246581 and the diagram of the transpose of (3,3,2,1) is
%e A246581    oooo
%e A246581    ooo
%e A246581    oo
%e A246581 and these diagrams share exactly (n-1)=8 cells in common, when the diagrams are positioned so that the upper-left corners of both diagrams coincide. (End)
%p A246581 fSp:=proc(k) local a,i,r;
%p A246581 a:=x^((k^2+k)/2)/mul(1-x^i,i=1..k);
%p A246581 a:=a/mul(1+x^r,r=1..101);
%p A246581 series(a,x,101);
%p A246581 seriestolist(%);
%p A246581 end;
%p A246581 fSp(2);
%t A246581 k = 2; CoefficientList[Series[x^((k^2 + k)/2)/(Product[1 - x^i, {i, k}] Product[1 + x^r, {r, 1000}]), {x, 0, 56}], x] (* _Michael De Vlieger_, Feb 01 2016 *)
%Y A246581 For k=0 and 1 we get A081362, A027349 (apart from signs).
%Y A246581 Cf. A218907, A246582, A246583.
%K A246581 sign
%O A246581 0,9
%A A246581 _N. J. A. Sloane_, Aug 31 2014