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 A242246 #11 May 17 2014 11:27:45 %S A242246 0,1,1,1,0,-1,0,1,0,-3,0,5,0,-691,0,35,0,-3617,0,43867,0,-1222277,0, %T A242246 854513,0,-1181820455,0,76977927,0,-23749461029,0,8615641276005,0, %U A242246 -84802531453387,0,90219075042845,0 %N A242246 Numerators of n*A164555(n-1)/A027642(n-1). %C A242246 First multiplied shifted (second) Bernoulli numbers. %C A242246 A164555(n-1)/A027642(n-1) = 0 followed by (A164555(n)/A027642(n)=1, 1/2, 1/6,...) = f(n) = 0, 1, 1/2, 1/6, 0,... . %C A242246 f(n+1) - f(n) = A051716(n)/A051717(n). %C A242246 Generally we consider a transform applied to the autosequences of first or second kind. An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. It is of the first kind if the main diagonal is A000004=0's. It is of the second kind if the main diagonal is the first upper diagonal multiplied by 2. A000045(n) is an autosequence of the first kind. A164555(n)/A027642(n) is an autosequence of the second kind. See A190339 (and A241269). %C A242246 Here we apply the transform to the Bernoulli numbers A164555(n)/A027642(n). %C A242246 We take n*(0 followed by A164555(n)/A027642(n)). %C A242246 Hence the autosequence of first kind %C A242246 TB1(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, O, -691/210,.. . %C A242246 a(n) are the numerators. %C A242246 The first seven rows of the differencece table of TB1(n) are %C A242246 0, 1, 1, 1/2, 0, - 1/6, 0, 1/6,... %C A242246 1, 0, -1/2, -1/2, -1/6, 1/6, 1/6, -1/6,... =A140351(n+1)/b(n+1) %C A242246 -1, -1/2, 0, 1/3, 1/3, 0, -1/3, -2/15,... %C A242246 1/2, 1/2, 1/3, 0, -1/3, -1/3, 1/5, 11/15,... %C A242246 0, -1/6, -1/3, -1/3, 0, 8/15, 8/15, -4/5,... %C A242246 -1/6, -1/6, 0, 1/3, 8/15, 0, -4/3, -4/3,... %C A242246 0, 1/6, 1/3, 1/5, -8/15, -4/3, 0, 512/105,... . %C A242246 First and second upper diagonals: 1, -1/2, 1/3, -1/3, 8/15, -4/3, 512/105,... . %C A242246 Sum of the antidiagonals: %C A242246 0, 1, 1, 0, -1/2, 0, 1/2, 0, -5/6, 0, 13/6, 0, -49/6, 0,... . %C A242246 (Note that the same transform applied to the second fractional Euler numbers A198631(n)/A006519(n+1) yields the Genocchi numbers -A226158(n)). %C A242246 This transform can be continued: %C A242246 TB2(n) = n*(0 followed by TB1(n)) = %C A242246 0, 0, 2, 3, 2, 0, -1, 0, 4/3, 0, -3, 0, 10, 0, -691/15, 0, 280, 0,... %C A242246 is an autosequence of second kind. %C A242246 TB3(n) = 0, 0, 0, 6, 12, 10, 0, -7, 0, 12, 0, -33, 0, 130, 0, 691, 0,... %C A242246 is apparently an integer autosequence of the first kind. %F A242246 a(n) = 0 followed by (A050925(n) = 1, -1, 1, 0,... ) with 1 instead of -1. %F A242246 a(2n) = A063524(n). a(2n+1) = A002427(n). %Y A242246 Cf. A199969 (autosequence). %K A242246 sign %O A242246 0,10 %A A242246 _Paul Curtz_, May 09 2014