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 A174690 #8 Feb 10 2021 01:43:18 %S A174690 1,1,1,1,3,1,1,13,13,1,1,73,121,73,1,1,481,1081,1081,481,1,1,3601, %T A174690 10081,13681,10081,3601,1,1,30241,100801,171361,171361,100801,30241,1, %U A174690 1,282241,1088641,2217601,2782081,2217601,1088641,282241,1,1,2903041,12700801,30119041,45360001,45360001,30119041,12700801,2903041,1 %N A174690 Triangle T(n, k) = n!*binomial(n, k) - n! + 1, read by rows. %H A174690 G. C. Greubel, <a href="/A174690/b174690.txt">Rows n = 0..100 of the triangle, flattened</a> %F A174690 T(n, k) = n!*binomial(n, k) - n! + 1. %F A174690 From _G. C. Greubel_, Feb 09 2021: (Start) %F A174690 T(n, k) = A196347(n, k) - n! + 1 = (-1)^k * A021012(n, k) - n! + 1. %F A174690 Sum_{k=0..n} T(n, k) = 2^n * n! - (n+1)! + (n+1) = A000165(n) - (n+1)! + (n+1). (End) %e A174690 Triangle begins as: %e A174690 1; %e A174690 1, 1; %e A174690 1, 3, 1; %e A174690 1, 13, 13, 1; %e A174690 1, 73, 121, 73, 1; %e A174690 1, 481, 1081, 1081, 481, 1; %e A174690 1, 3601, 10081, 13681, 10081, 3601, 1; %e A174690 1, 30241, 100801, 171361, 171361, 100801, 30241, 1; %e A174690 1, 282241, 1088641, 2217601, 2782081, 2217601, 1088641, 282241, 1; %t A174690 T[n_, k_]:= n!*Binomial[n, k] - n! + 1; %t A174690 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten %o A174690 (Sage) flatten([[factorial(n)*(binomial(n,k) -1) + 1 for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 09 2021 %o A174690 (Magma) [Factorial(n)*(Binomial(n,k) -1) + 1: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 09 2021 %Y A174690 Cf. A000165, A021012, A196347. %K A174690 nonn,tabl,easy %O A174690 0,5 %A A174690 _Roger L. Bagula_, Mar 27 2010 %E A174690 Edited by _G. C. Greubel_, Feb 09 2021