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A378704 Array read by ascending antidiagonals: A(n, k) is the total area of n-Fibonacci polyominoes with k columns, where k > 0.

%I A378704 #13 Dec 06 2024 11:30:32
%S A378704 2,3,7,4,11,16,5,15,31,35,6,19,43,73,70,7,23,55,111,168,136,8,27,67,
%T A378704 143,261,370,256,9,31,79,175,351,602,790,473,10,35,91,207,431,816,
%U A378704 1350,1658,860,11,39,103,239,511,1023,1865,2966,3425,1545,12,43,115,271,591,1215,2346,4178,6414,6989,2748
%N A378704 Array read by ascending antidiagonals: A(n, k) is the total area of n-Fibonacci polyominoes with k columns, where k > 0.
%H A378704 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 pages 8-9.
%F A378704 A(n, k) = [y^k] y*(n^2*(1 - y)^2*y^n + 2*y*(1 - y^n) - n(1 - y)*(2- y^n + y^(n+1)))/(2*(-1 + y)*(1 - 2*y + y^(n+1))^2).
%F A378704 A(n, n-1) = A356888(n) - 1.
%e A378704 The array begins as:
%e A378704   2,  7, 16,  35,  70,  136,  256, ...
%e A378704   3, 11, 31,  73, 168,  370,  790, ...
%e A378704   4, 15, 43, 111, 261,  602, 1350, ...
%e A378704   5, 19, 55, 143, 351,  816, 1865, ...
%e A378704   6, 23, 67, 175, 431, 1023, 2346, ...
%e A378704   7, 27, 79, 207, 511, 1215, 2815, ...
%e A378704   ...
%t A378704 A[n_,k_]:=SeriesCoefficient[y(n^2(1-y)^2y^n+2y(1-y^n)-n(1-y)(2-y^n+y^(n+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 A378704 Cf. A004767 (k=2), A017605 (k=3), A356888, A378706, A378707, A378716.
%K A378704 nonn,tabl
%O A378704 2,1
%A A378704 _Stefano Spezia_, Dec 04 2024