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 A241209 #48 Jun 08 2023 08:51:03
%S A241209 1,1,-1,-5,5,61,-61,-1385,1385,50521,-50521,-2702765,2702765,
%T A241209 199360981,-199360981,-19391512145,19391512145,2404879675441,
%U A241209 -2404879675441,-370371188237525,370371188237525,69348874393137901,-69348874393137901,-15514534163557086905
%N A241209 a(n) = E(n) - E(n+1), where E(n) are the Euler numbers A122045(n).
%C A241209 A version of the Seidel triangle (1877) for the integer Euler numbers is
%C A241209 1
%C A241209 1 1
%C A241209 2 2 1
%C A241209 2 4 5 5
%C A241209 16 16 14 10 5
%C A241209 16 32 46 56 61 61
%C A241209 etc.
%C A241209 It is not in the OEIS. See A008282.
%C A241209 The first diagonal, Es(n) = 1, 1, 1, 5, 5, 61, 61, 1385, 1385, ..., comes from essentially A000364(n) repeated.
%C A241209 a(n) is Es(n) signed two by two.
%C A241209 Difference table of a(n):
%C A241209 1, 1, -1, -5, 5, 61, -61, -1385, ...
%C A241209 0, -2, -4, 10, 56, -122, -1324, ...
%C A241209 -2, -2, 14, 46, -178, -1202, ...
%C A241209 0, 16, 32, -224, -1024, ...
%C A241209 16, 16, -256, -800, ...
%C A241209 0, -272, -544, ...
%C A241209 -272, -272, ...
%C A241209 0, ...
%C A241209 etc.
%C A241209 Sum of the antidiagonals: 1, 1, -5, -11, 61, 211, -385, ... = A239322(n+1).
%C A241209 Main diagonal interleaved with the first upper diagonal: 1, 1, -2, -4, 14, 46, ... = signed A214267(n+1).
%C A241209 Inverse binomial transform (first column): A155585(n+1).
%C A241209 The Akiyama-Tanigawa transform applied to A046978(n+1)/A016116(n) gives
%C A241209 1, 1, 1/2, 0, -1/4, -1/4, -1/8, 0, ...
%C A241209 0, 1, 3/2, 1, 0, -3/4, -7/8, ...
%C A241209 -1, -1, 3/2, 4, 15/4, 3/4, ...
%C A241209 0, -5, -15/2, 1, 15, ...
%C A241209 5, 5, -51/2, -56, ...
%C A241209 0, 61, 183/2, ...
%C A241209 -61, -61, ...
%C A241209 0, ...
%C A241209 etc.
%C A241209 A122045(n) and A239005(n) are reciprocal sequences by their inverse binomial transform. In their respective difference table, two different signed versions of A214247(n) appear: 1) interleaved main diagonal and first under diagonal (1, -1, -1, 2, 4, -14, ...) and 2) interleaved main diagonal and first upper diagonal (1, 1, -1, -2, 4, 14, ...).
%H A241209 G. C. Greubel, <a href="/A241209/b241209.txt">Table of n, a(n) for n = 0..475</a>
%F A241209 a(n) = A119880(n+1) - A119880(n).
%F A241209 a(n) is the second column of the fractional array.
%F A241209 a(n) = (-1)^n*second column of the array in A239005(n).
%F A241209 a(n) = skp(n, 0) - skp(n+1, 0), where skp(n, x) are the Swiss-Knife polynomials A153641. - _Peter Luschny_, Apr 17 2014
%F A241209 E.g.f.: exp(x)/cosh(x)^2. - _Sergei N. Gladkovskii_, Jan 23 2016
%F A241209 G.f. T(0)/x-1/x, where T(k) = 1 - x*(k+1)/(x*(k+1)-(1-x)/(1-x*(k+1)/(x*(k+1)+(1-x)/T(k+1)))). - _Sergei N. Gladkovskii_, Jan 23 2016
%p A241209 A241209 := proc(n) local v, k, h, m; m := `if`(n mod 2 = 0, n, n+1);
%p A241209 h := k -> `if`(k mod 4 = 0, 0, (-1)^iquo(k,4));
%p A241209 (-1)^n*add(2^iquo(-k,2)*h(k+1)*add((-1)^v*binomial(k,v)*(v+1)^m, v=0..k)
%p A241209 ,k=0..m) end: seq(A241209(n),n=0..24); # _Peter Luschny_, Apr 17 2014
%t A241209 skp[n_, x_]:= Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}];
%t A241209 a[n_]:= skp[n, x] - skp[n+1, x]/. x->0; Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Apr 17 2014, after _Peter Luschny_ *)
%t A241209 Table[EulerE[n] - EulerE[n+1], {n,0,30}] (* _Vincenzo Librandi_, Jan 24 2016 *)
%t A241209 -Differences/@Partition[EulerE[Range[0,30]],2,1]//Flatten (* _Harvey P. Dale_, Apr 16 2019 *)
%o A241209 (Magma)
%o A241209 EulerPoly:= func< n,x | (&+[ (&+[ (-1)^j*Binomial(k,j)*(x+j)^n : j in [0..k]])/2^k: k in [0..n]]) >;
%o A241209 Euler:= func< n | 2^n*EulerPoly(n, 1/2) >; // A122045
%o A241209 [Euler(n) - Euler(n+1): n in [0..40]]; // _G. C. Greubel_, Jun 07 2023
%o A241209 (SageMath) [euler_number(n) - euler_number(n+1) for n in range(41)] # _G. C. Greubel_, Jun 07 2023
%Y A241209 Cf. A000364, A000657, A008282, A016116, A046978, A119880, A122045, A153641, A155585, A214247, A214267, A239005, A239322.
%K A241209 sign
%O A241209 0,4
%A A241209 _Paul Curtz_, Apr 17 2014