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A217874 Table A142978 (figurate numbers for n-dimensional cross polytope) extended by a top row.

%I A217874 #16 Jan 03 2025 23:28:21
%S A217874 1,1,0,1,2,1,1,4,3,0,1,6,9,4,1,1,8,19,16,5,0,1,10,33,44,25,6,1,1,12,
%T A217874 51,96,85,36,7,0,1,14,73,180,225,146,49,8,1,1,16,99,304,501,456,231,
%U A217874 64,9,0,1,18,129,476,985,1182,833,344,81,10,1
%N A217874 Table A142978 (figurate numbers for n-dimensional cross polytope) extended by a top row.
%C A217874 Looking at table A142978, it seems natural to extend it by a first row, which turns out to be 1,0,1,0,1,0,.... Indeed, these are the values obtained when the polynomial which defines each of the columns, is "extrapolated" to n=0.
%H A217874 J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, <a href="https://doi.org/10.37236/3478">Tiling a strip with triangles</a>, El. J. Combinat. 21 (1) (2014) P1.7.
%e A217874 The table may be written as:
%e A217874   1,_ v---- the even numbers A005843
%e A217874   1,_ 0,__ v---- A058331(n) = 2*n^2 + 1.
%e A217874   1,_ 2,__ 1,__ v---- 4*A006527(n) = 4n(n^2 + 2)/3.
%e A217874   1,_ 4,__ 3,__ 0,__ v---- 2n^2(n^2 + 5)/3 + 1.
%e A217874   1,_ 6,__ 9,__ 4,__ 1,___ v---- 2n(2n^4 + 20n^2 + 23)/15.
%e A217874   1,_ 8,_ 19,_ 16,__ 5,___ 0,
%e A217874   1, 10,_ 33,_ 44,_ 25,___ 6,__ 1,
%e A217874   1, 12,_ 51,_ 96,_ 85,__ 36,__ 7,__ 0,
%e A217874   1, 14,_ 73, 180, 225,_ 146,_ 49,__ 8,_ 1,
%e A217874   1, 16,_ 99, 304, 501,_ 456, 231,_ 64,_ 9, 0,
%e A217874   1, 18, 129, 476, 985, 1182, 833, 344, 81, 10, 1,...
%e A217874 Sequence A142978 is the table obtained by deleting the uppermost row 1,0,1,0,1,...
%e A217874 (One could also add a 0th column, with all zeros.)
%K A217874 nonn,tabl
%O A217874 0,5
%A A217874 _M. F. Hasler_, Oct 13 2012