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 A128139 #19 Feb 15 2022 11:10:37
%S A128139 1,1,2,1,3,3,1,4,5,4,1,5,7,7,5,1,6,9,10,9,6,1,7,11,13,13,11,7,1,8,13,
%T A128139 16,17,16,13,8,1,9,15,19,21,21,19,15,9,1,10,17,22,25,26,25,22,17,10
%N A128139 Triangle read by rows: matrix product A004736 * A128132.
%C A128139 A077028 with the final term in each row omitted.
%C A128139 Interchanging the factors in the matrix product leads to A128140 = A128132 * A004736.
%C A128139 From _Gary W. Adamson_, Jul 01 2012: (Start)
%C A128139 Alternatively, antidiagonals of an array A(n,k) of sequences with arithmetic progressions as follows:
%C A128139 1, 2, 3, 4, 5, 6, ...
%C A128139 1, 3, 5, 7, 9, 11, ...
%C A128139 1, 4, 7, 10, 13, 16, ...
%C A128139 1, 5, 9, 13, 17, 21, ...
%C A128139 ... (End)
%C A128139 From _Gary W. Adamson_, Jul 02 2012: (Start)
%C A128139 A summation generalization for Sum_{k>=1} 1/(A(n,k)*A(n,k+1)) (formulas copied from A002378, A000466, A085001, A003185):
%C A128139 1 = 1/(1)*(2) + 1/(2)*(3) + 1/(3)*(4) + ...
%C A128139 1 = 2/(1)*(3) + 2/(3)*(5) + 2/(5)*(7) + ...
%C A128139 1 = 3/(1)*(4) + 3/(4)*(7) + 3/(7)*(10) + ...
%C A128139 1 = 4/(1)*(5) + 4/(5)*(9) + 4/(9)*(13) + ...
%C A128139 ...
%C A128139 As a summation of terms equating to a definite integral:
%C A128139 Integral_{0..1} dx/(1+x) = ... 1 - 1/2 + 1/3 - 1/4 + ... = log(2).
%C A128139 Integral_{0..1} dx/(1+x^2) = 1 - 1/3 + 1/5 - 1/7 + ... = Pi/4 (see A157142)
%C A128139 Integral_{0..1} dx/(1+x^3) = 1 - 1/4 + 1/7 - 1/10 + ... (see A016777)
%C A128139 Integral_{0..1} dx/(1+x^4) = 1 - 1/5 + 1/9 - 1/13 + ... (see A016813). (End)
%F A128139 A004736 * A128132 as infinite lower triangular matrices.
%F A128139 T(n,k) = k*(1+n-k)+1 = 1 + A094053(n+1,1+n-k). - _R. J. Mathar_, Jul 09 2012
%e A128139 First few rows of the triangle:
%e A128139 1;
%e A128139 1, 2;
%e A128139 1, 3, 3;
%e A128139 1, 4, 5, 4;
%e A128139 1, 5, 7, 7, 5;
%e A128139 1, 6, 9, 10, 9, 6;
%e A128139 1, 7, 11, 13, 13, 11, 7;
%e A128139 1, 8, 13, 16, 17, 16, 13, 8;
%e A128139 1, 9, 15, 19, 21, 21, 19, 15, 9;
%e A128139 1, 10, 17, 22, 25, 26, 25, 22, 17, 10;
%e A128139 ...
%Y A128139 Cf. A004736, A128132, A128140, A004006 (row sums).
%K A128139 nonn,easy,tabl
%O A128139 0,3
%A A128139 _Gary W. Adamson_, Feb 16 2007