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%I #9 Mar 30 2024 23:08:41
%S 1,1,2,1,2,5,1,2,2,16,1,2,2,5,45,1,2,2,2,12,125,1,2,2,2,5,24,341,1,2,
%T 2,2,2,12,48,918,1,2,2,2,2,7,18,97,2453,1,2,2,2,2,2,16,28,195,6515
%N Triangle read by rows, antidiagonals of an array generated from INVERT transforms of variants of (1, 2, 3, ...).
%C Row sums = A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, ...).
%F Given S(x) = (1 + 2x + 3x^2 + ...), where (1, 2, 3, ...) = the INVERTi transform of (1, 3, 8, 21, 55, ...); k-th row of the array = INVERT transform of S(x^k). Take finite differences of array columns starting from the topmost "1"; becoming rows of the triangle.
%e First few rows of the array:
%e 1, 3, 8, 21, 55, 144, 377, 987, 2584, ...
%e 1, 1, 3, 5, 10, 19, 36, 69, 131, ...
%e 1, 1, 1, 3, 5, 7, 12, 21, 34, ...
%e 1, 1, 1, 1, 3, 5, 7, 9, 16, ...
%e 1, 1, 1, 1, 1, 3, 5, 7, 9, ...
%e 1, 1, 1, 1, 1, 1, 3, 5, 7, ...
%e ...
%e Taking finite differences from the bottom to top starting with the last "1" we obtain triangle A175011:
%e 1;
%e 1, 2;
%e 1, 2, 5;
%e 1, 2, 2, 16;
%e 1, 2, 2, 5, 45;
%e 1, 2, 2, 2, 12, 125;
%e 1, 2, 2, 2, 5, 24, 341;
%e 1, 2, 2, 2, 2, 12, 48, 918;
%e 1, 2, 2, 2, 2, 7, 18, 97, 2453;
%e 1, 2, 2, 2, 2, 2, 16, 28, 195, 6515;
%e ...
%Y Cf. A001906.
%K nonn,tabl
%O 1,3
%A _Gary W. Adamson_, Apr 03 2010
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