This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346874 #34 Aug 26 2021 15:18:01 %S A346874 1,2,1,4,2,1,8,3,2,1,1,16,6,3,2,2,1,1,32,11,6,4,2,2,2,1,2,1,64,22,11, %T A346874 7,5,3,3,2,2,2,1,2,1,1,1,128,43,22,13,9,7,5,4,3,3,3,2,2,1,2,1,2,1,1,1, %U A346874 1,1,256,86,43,26,18,12,10,8,6,5,4,4,3,3,3,2,3 %N A346874 Irregular triangle read by rows in which row n lists the row 2^n - 1 of A237591, n >= 1. %C A346874 The Mersenne number A000225(n) does not has a characteristic shape of its symmetric representation of sigma(A000225(n)). On the other hand, we can find that number in two ways in the symmetric representation of the powers of 2 as follows: the Mersenne numbers are the semilength of the smallest Dyck path and also they equals the area (or the number of cells) of the region of the diagram (see examples). %C A346874 Therefore we can see a geometric pattern of the distribution of the Mersenne numbers in the stepped pyramid described in A245092. %C A346874 T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000225(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000225(n). %C A346874 T(n,k) is also the difference between the total number of partitions of all positive integers <= Mersenne number A000225(n) into k consecutive parts, and the total number of partitions of all positive integers <= Mersenne number A000225(n) into k + 1 consecutive parts. %e A346874 Triangle begins: %e A346874 1; %e A346874 2, 1; %e A346874 4, 2, 1; %e A346874 8, 3, 2, 1, 1; %e A346874 16, 6, 3, 2, 2, 1, 1; %e A346874 32, 11, 6, 4, 2, 2, 2, 1, 2, 1; %e A346874 64, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1; %e A346874 128, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1; %e A346874 ... %e A346874 Illustration of initial terms: %e A346874 . %e A346874 Row 1: %e A346874 0_ Semilength = 0 Area = 1 %e A346874 |_| %e A346874 Row 2: %e A346874 _ %e A346874 1_| | Semilength = 1 Area = 3 %e A346874 |_ _| %e A346874 . %e A346874 Row 3: _ %e A346874 | | %e A346874 1 _| | %e A346874 2_ _| _| Semilength = 3 Area = 7 %e A346874 |_ _ _| %e A346874 . %e A346874 Row 4: _ %e A346874 | | %e A346874 | | %e A346874 | | %e A346874 _ _| | %e A346874 1 _| _ _| %e A346874 4 2| _| Semilength = 7 Area = 15 %e A346874 _ _ _ _| | %e A346874 |_ _ _ _ _| %e A346874 . %e A346874 Row 5: _ %e A346874 | | %e A346874 | | %e A346874 | | %e A346874 | | %e A346874 | | %e A346874 | | %e A346874 | | %e A346874 _ _ _| | %e A346874 | _ _ _| %e A346874 _| | %e A346874 1 1_| _| %e A346874 2 _ _| _| Semilength = 15 Area = 31 %e A346874 | _ _| %e A346874 8 3| | %e A346874 _ _ _ _ _ _ _ _| | %e A346874 |_ _ _ _ _ _ _ _ _| %e A346874 . %Y A346874 Row sums give A000225, n >= 1. %Y A346874 Column 1 gives A000079. %Y A346874 For the characteristic shape of sigma(A000040(n)) see A346871. %Y A346874 For the characteristic shape of sigma(A000079(n)) see A346872. %Y A346874 For the characteristic shape of sigma(A000217(n)) see A346873. %Y A346874 For the characteristic shape of sigma(A000384(n)) see A346875. %Y A346874 For the characteristic shape of sigma(A000396(n)) see A346876. %Y A346874 For the characteristic shape of sigma(A008588(n)) see A224613. %Y A346874 Cf. A000203, A237591, A237593, A245092, A249351, A262626. %K A346874 nonn,tabf %O A346874 1,2 %A A346874 _Omar E. Pol_, Aug 06 2021