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A378707 Array read by ascending antidiagonals: A(n,k) is the total number of inner points of n-Fibonacci polyominoes with k columns, where k > 0.

%I A378707 #13 Dec 06 2024 11:11:36
%S A378707 0,0,1,0,3,3,0,5,10,7,0,7,18,26,15,0,9,26,50,63,30,0,11,34,74,124,143,
%T A378707 58,0,13,42,98,190,296,313,109,0,15,50,122,254,457,679,668,201,0,17,
%U A378707 58,146,318,622,1070,1517,1398,365,0,19,66,170,382,782,1461,2439,3325,2883,655
%N A378707 Array read by ascending antidiagonals: A(n,k) is the total number of inner points of n-Fibonacci polyominoes with k columns, where k > 0.
%H A378707 Juan F. Pulido, José L. Ramírez, and Andrés R. Vindas-Meléndez, <a href="https://arxiv.org/abs/2411.17812">Generating Trees and Fibonacci Polyominoes</a>, arXiv:2411.17812 [math.CO], 2024. See page 12.
%F A378707 A(n, k) = [y^k] y*((6 - 4*n)*y - (2 - 4*n)*y^2 - (3 - n)*n*y^n -2*(2 - n)^2*y^(n+1) + (2 - 5*n + n^2)*y^(n+2) + 2*y^(2n+1))/(2*(-1 + y)*(1 - 2*y + y^(n+1))^2).
%F A378707 A(2, n) = A023610(n-2) for n > 1.
%e A378707 The array begins as:
%e A378707   0,  1,  3,   7,  15,  30,   58,  109,   201,   365, ...
%e A378707   0,  3, 10,  26,  63, 143,  313,  668,  1398,  2883, ...
%e A378707   0,  5, 18,  50, 124, 296,  679, 1517,  3325,  7184, ...
%e A378707   0,  7, 26,  74, 190, 457, 1070, 2439,  5453, 12013, ...
%e A378707   0,  9, 34,  98, 254, 622, 1461, 3361,  7583, 16857, ...
%e A378707   0, 11, 42, 122, 318, 782, 1854, 4272,  9681, 21615, ...
%e A378707   0, 13, 50, 146, 382, 942, 2238, 5182, 11754, 26302, ...
%e A378707   ...
%t A378707 A[n_, k_]:=SeriesCoefficient[y((6-4n)y-(2-4n)y^2-(3-n)n y^n-2(2-n)^2y^(n+1)+(2-5n+n^2)y^(n+2)+2y^(2n+1))/(2(-1+y)(1-2y+y^(n+1))^2), {y, 0, k}]; Table[A[n-k+1, k], {n, 2, 12}, {k, n-1}]//Flatten
%Y A378707 Cf. A023610, A378704, A378706, A378716.
%K A378707 nonn,tabl
%O A378707 2,5
%A A378707 _Stefano Spezia_, Dec 05 2024