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 A130103 #45 Mar 02 2025 23:41:45
%S A130103 0,1,1,4,11,26,57,120,247,502,1013,2036,4083,8178,16369,32752,65519,
%T A130103 131054,262125,524268,1048555,2097130,4194281,8388584,16777191,
%U A130103 33554406,67108837,134217700,268435427,536870882,1073741793,2147483616
%N A130103 Expansion of e.g.f. e^(2x)-(1+x)*e^x+x.
%C A130103 Partial sums are A130104.
%C A130103 Essentially the same as the Euler numbers A000295.
%C A130103 Number of binary strings of length n where 0 is not used or is used at least twice and 1 is used at least once. For example, for n=3 the strings are 100, 010, 001, 111. - _Enrique Navarrete_, Feb 06 2025
%C A130103 Also the number of ordered set partitions of an n-set into 2 sets such that the first set cannot have a single element and the second set has at least one element. For example, for n=3 the ordered set partitions are: { },{1,2,3}; {1,2},{3}; {1,3},{2}; {2,3},{1}. - _Enrique Navarrete_, Feb 14 2025
%H A130103 Seiichi Manyama, <a href="/A130103/b130103.txt">Table of n, a(n) for n = 0..3321</a>
%H A130103 Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, <a href="https://arxiv.org/abs/2501.06675">Chip-Firing on Infinite k-ary Trees</a>, arXiv:2501.06675 [math.CO], 2025. See p. 18.
%H A130103 Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%H A130103 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,2).
%F A130103 G.f.: x(1-3x+5x^2-2x^3)/((1-x)^2*(1-2x)).
%F A130103 E.g.f.: e^(2x)-(1+x)*e^x+x.
%F A130103 a(n) = 2^n-n-1+C(1,n)-C(0,n).
%F A130103 a(n) = A130102(n+1)/2.
%F A130103 a(n) = Sum_{i=1..n} i*2^(n-i) - _Ctibor O. Zizka_, Feb 23 2008
%e A130103 G.f. = x + x^2 + 4*x^3 + 11*x^4 + 26*x^5 + 57*x^6 + 120*x^7 + 247*x^8 + ...
%p A130103 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+n od: seq(a[n], n=0..30); # _Zerinvary Lajos_, Feb 22 2008
%t A130103 Join[{0,1},LinearRecurrence[{4,-5,2},{1,4,11},40]] (* _Harvey P. Dale_, May 16 2014 *)
%t A130103 a[ n_] := If[ n < 2, Boole[n == 1], 2^n - (1 + n)]; (* _Michael Somos_, Aug 17 2015 *)
%o A130103 (PARI) {a(n) = if( n<2, n==1, 2^n - (1+n))}; /* _Michael Somos_, Aug 17 2015 */
%Y A130103 Cf. A000295.
%K A130103 easy,nonn
%O A130103 0,4
%A A130103 _Paul Barry_, May 07 2007