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 A252117 #61 Oct 29 2022 04:49:50
%S A252117 1,3,9,5,22,15,51,45,108,110,14,221,255,42,429,540,126,810,1105,308,
%T A252117 1479,2145,714,30,2640,4050,1512,90,4599,7395,3094,270,7868,13200,
%U A252117 6006,660,13209,22995,11340,1530,21843,39340,20706,3240,55,35581,66045,36960,6630,165,57222,109215,64386,12870,495
%N A252117 Irregular triangle read by row: T(n,k), n>=1, k>=1, in which column k lists the numbers of A000716 multiplied by A000330(k), and the first element of column k is in row A000217(k).
%C A252117 Gives an identity for sigma(n). Alternating sum of row n equals A000203(n), the sum of the divisors of n.
%C A252117 Row n has length A003056(n) hence column k starts in row A000217(k).
%C A252117 Column 1 is A000716, but here the offset is 1 not 0.
%C A252117 The 1st element of column k is A000330(k).
%C A252117 The 2nd element of column k is A059270(k).
%C A252117 The 3rd element of column k is A220443(k).
%C A252117 The partial sums of column k give the k-th column of A249120.
%C A252117 This triangle has been constructed after _Mircea Merca_'s formula for A000203.
%C A252117 From _Omar E. Pol_, May 05 2022: (Start)
%C A252117 In the Honda-Yoda paper see "3. String theory and Riemann hypothesis". The coefficients that are mentioned in 3.11 are the first 16 terms of A000716, the coefficients that are mentioned in 3.12 are the first 5 terms of A000330, and the coefficients that are mentioned in 3.13 are the first 16 terms of A000203.
%C A252117 Another triangle with the same row lengths and whose alternating row sums give A000203 is A196020. (End)
%H A252117 Masazumi Honda and Takuya Yoda, <a href="https://arxiv.org/abs/2203.17091">String theory, N = 4 SYM and Riemann hypothesis</a>, arXiv:2203.17091 [hep-th], (2022), pp. 5-6.
%H A252117 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A252117 A000203(n) = Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k).
%e A252117 Triangle begins:
%e A252117 1;
%e A252117 3;
%e A252117 9, 5;
%e A252117 22, 15;
%e A252117 51, 45;
%e A252117 108, 110, 14;
%e A252117 221, 255, 42;
%e A252117 429, 540, 126;
%e A252117 810, 1105, 308;
%e A252117 1479, 2145, 714, 30;
%e A252117 2640, 4050, 1512, 90;
%e A252117 4599, 7395, 3094, 270;
%e A252117 7868, 13200, 6006, 660;
%e A252117 13209, 22995, 11340, 1530;
%e A252117 21843, 39340, 20706, 3240, 55;
%e A252117 35581, 66045, 36960, 6630, 165;
%e A252117 57222, 109215, 64386, 12870, 495;
%e A252117 90882, 177905, 110152, 24300, 1210;
%e A252117 142769, 286110, 184926, 44370, 2805;
%e A252117 221910, 454410, 305802, 79200, 5940;
%e A252117 341649, 713845, 498134, 137970, 12155, 91;
%e A252117 ...
%e A252117 For n = 6 the divisors of 6 are 1, 2, 3, 6, so the sum of the divisors of 6 is 1 + 2 + 3 + 6 = 12. On the other hand, the 6th row of the triangle is 108, 110, 14, so the alternating row sum is 108 - 110 + 14 = 12, equaling the sum of the divisors of 6.
%e A252117 For n = 15 the divisors of 15 are 1, 3, 5, 15, so the sum of the divisors of 15 is 1 + 3 + 5 + 15 = 24. On the other hand, the 15th row of the triangle is 21843, 39340, 20706, 3240, 55, so the alternating row sum is 21843 - 39340 + 20706 - 3240 + 55 = 24, equaling the sum of the divisors of 15.
%o A252117 (PARI) A003056(n) = (sqrtint(8*n+1)-1)\2;
%o A252117 A000330(n) = n*(n+1)*(2*n+1)/6;
%o A252117 A000217(n) = n*(n+1)/2;
%o A252117 A000716(n) = polcoef(1/eta('x+O('x^(n+1)))^3, n, x);
%o A252117 T(n, k) = A000330(k)*A000716(n-A000217(k));
%o A252117 row(n) = vector(A003056(n), k, T(n,k)); \\ _Michel Marcus_, Oct 29 2022
%Y A252117 Cf. A000203, A000217, A000330, A000716, A003056, A059270, A196020, A220443, A238442, A249120.
%K A252117 nonn,tabf
%O A252117 1,2
%A A252117 _Omar E. Pol_, Dec 14 2014