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Irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order.
109

%I #21 Jan 17 2023 21:24:56

%S 1,2,2,1,3,3,1,3,2,3,2,1,4,4,1,4,2,4,2,1,4,3,4,3,1,4,3,2,4,3,2,1,5,5,

%T 1,5,2,5,2,1,5,3,5,3,1,5,3,2,5,3,2,1,5,4,5,4,1,5,4,2,5,4,2,1,5,4,3,5,

%U 4,3,1,5,4,3,2,5,4,3,2,1,6,6,1,6,2,6,2

%N Irregular triangle read by rows: strictly decreasing sequences of positive numbers given in lexicographic order.

%C Length of n-th row given by A000120(n);

%C Min of n-th row given by A001511(n);

%C Sum of n-th row given by A029931(n);

%C Product of n-th row given by A096111(n);

%C Max of n-th row given by A113473(n);

%C Numerator of sum of reciprocals of n-th row given by A116416(n);

%C Denominator of sum of reciprocals of n-th row given by A116417(n);

%C LCM of n-th row given by A271410(n).

%C The first appearance of n is at A001787(n - 1).

%C n-th row begins at index A000788(n - 1) for n > 0.

%C Also the reversed positions of 1's in the reversed binary expansion of n. Also the reversed partial sums of the n-th composition in standard order (row n of A066099). Reversing rows gives A048793. - _Gus Wiseman_, Jan 17 2023

%H Peter Kagey, <a href="/A272020/b272020.txt">Table of n, a(n) for n = 0..10000</a>

%e Row n is given by the exponents in the binary expansion of 2*n. For example, row 5 = [3, 1] because 2*5 = 2^3 + 2^1.

%e Row 0: []

%e Row 1: [1]

%e Row 2: [2]

%e Row 3: [2, 1]

%e Row 4: [3]

%e Row 5: [3, 1]

%e Row 6: [3, 2]

%e Row 7: [3, 2, 1]

%t Table[Reverse[Join@@Position[Reverse[IntegerDigits[n,2]],1]],{n,0,100}] (* _Gus Wiseman_, Jan 17 2023 *)

%Y Cf. A000120, A096111, A116416, A116417, A271410, A001787.

%Y Cf. A048793 gives the rows in reverse order.

%Y Cf. A272011.

%Y Lasts are A001511.

%Y Heinz numbers of the rows are A019565.

%Y Firsts are A029837 or A070939 or A113473.

%Y Row sums are A029931.

%Y A066099 lists standard comps, partial sums A358134, weighted sum A359042.

%Y Cf. A005940, A059893, A125106, A161511, A242628.

%K nonn,tabf

%O 0,2

%A _Peter Kagey_, Apr 17 2016