A398028 - OEIS

%I #7 Jul 20 2026 09:35:31

%S 0,0,1,2,7,28,120,528,2344,10416,46160,203676,894228,3906036,16976952,

%T 73439268,316282360,1356560728,5796404302,24681075816,104755646898,

%U 443313571324,1870972607370,7876620943924,33083632385428,138664016164056,580044174368274,2421965007900888,10095807143173876

%N Number of 7-convex polyominoes with semiperimeter n.

%H Andrew R. Conway and Anthony J. Guttmann, <a href="https://arxiv.org/abs/2606.13845">k-Convex Polyominoes by Semi-perimeter</a>, arXiv:2606.13845 [cs.DM], Jul 17 2026.

%F G.f.: x^2*(24*x^10 - 198*x^9 + 867*x^8 - 2476*x^7 + 4072*x^6 - 3896*x^5 + 2251*x^4 - 798*x^3 + 170*x^2 - 20*x + 1) / ((1-2*x) * (1-4*x)^2 * (2*x^2 - 4*x + 1) * (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)) - x^4*(7*x^3 - 14*x^2 + 7*x - 1)^2 / ((1-4*x)^(3/2) * (1-2*x) * (2*x^2 - 4*x + 1) * (2*x^4 - 16*x^3 + 20*x^2 - 8*x + 1)) [Conway and Guttmann, 2026].

%F For asymptotics see Conway and Guttmann (2026).

%t CoefficientList[Series[(-2*x^4 + x^2*(1 - 6*x + 7*x^2 + 4*x^3) * ChebyshevU[k, (1-2*x)/(2*x)] + x^3*(1 - 4*x + 8*x^2) * ChebyshevU[k-1, (1-2*x)/(2*x)]) / ((1-4*x)^2 * ChebyshevU[k, (1-2*x)/(2*x)]) - 4*x^4 * (ChebyshevT[k, 1/(2*Sqrt[x])]^2/((1 - 4*x)^(3/2) * ChebyshevU[k, (1-2*x)/(2*x)])) /. k->7, {x, 0, 28}], x]

%Y Cf. A128611, A398024, A398025, A398026, A398027.

%K nonn,new

%O 0,4

%A _Vaclav Kotesovec_, Jul 17 2026