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 A045618 #77 May 12 2025 10:00:17
%S A045618 1,6,23,72,201,522,1291,3084,7181,16398,36879,81936,180241,393234,
%T A045618 851987,1835028,3932181,8388630,17825815,37748760,79691801,167772186,
%U A045618 352321563,738197532,1543503901,3221225502,6710886431,13958643744
%N A045618 Partial sums of A000337(n+4), n >= 0.
%C A045618 Convolution of A000225(n+1), n >= 0, (partial sums of powers of 2).
%C A045618 Sum of diameters of all nonempty subsets of {1, 2, ..., n+2}. - _Charles R Greathouse IV_, Nov 21 2011
%C A045618 a(n) is the sum of all the ways of adding the k-tuples of the terms found in A000079(0) to A000079(n). For a(2) the result is (1)+(2)+(4)=7; (1+2)+(2+4)=9; (1+2+4)=7 with 7+9+7=23. - _J. M. Bergot_, Jun 19 2017
%H A045618 Vincenzo Librandi, <a href="/A045618/b045618.txt">Table of n, a(n) for n = 0..1000</a>
%H A045618 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. 14.
%H A045618 Brian Nakamura and Elizabeth Yang, <a href="http://arxiv.org/abs/1503.05617">Competition graphs induced by permutations</a>, arXiv preprint arXiv:1503.05617 [math.CO], 2015.
%H A045618 Alina F. Y. Zhao, <a href="http://www.emis.de/journals/JIS/VOL17/Zhao/zhao3.html">Pattern Popularity in Multiply Restricted Permutations</a>, Journal of Integer Sequences, 17 (2014), #14.10.3.
%H A045618 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).
%F A045618 a(n) = n + 5 + (n-1)*2^(n+2).
%F A045618 G.f.: 1/((1-2*x)*(1-x))^2.
%F A045618 a(n) = Sum_{i=0...n+1} (2^(n+2-i) - 1)*(2^i - 1). - _J. M. Bergot_, Sep 16 2017
%F A045618 a(n) = Sum_{k=0..n+2} Sum_{i=0..n+2} (i-k) * C(n-k+2,i). - _Wesley Ivan Hurt_, Sep 19 2017
%F A045618 a(n) = 4*a(n-1) - 4*a(n-2) + n + 1, with a(-1) = a(-2) = 0. - _Jesse Fiedler_, Aug 20 2019
%F A045618 E.g.f.: exp(x)*(5 + x + exp(x)*(- 4 + 8*x)). - _Stefano Spezia_, Aug 20 2019
%F A045618 a(n) = Sum_{k=0..n} Stirling2(k+2,2) * Stirling2(n-k+2,2). - _Seiichi Manyama_, May 12 2025
%t A045618 Table[Sum[(-1)^(n - k) k (-1)^(n - k) Binomial[n + 2, k + 2], {k, 0, n}], {n, 1, 28}] (* _Zerinvary Lajos_, Jul 08 2009 *)
%t A045618 Rest[Accumulate[LinearRecurrence[{5,-8,4},{0,1,5},40]]] (* _Harvey P. Dale_, Dec 19 2011 *)
%t A045618 CoefficientList[Series[1/((1 - x)^2 (1 - 2 x)^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Jul 22 2014 *)
%t A045618 LinearRecurrence[{6, -13, 12, -4},{1, 6, 23, 72},28] (* _Ray Chandler_, Aug 03 2015 *)
%o A045618 (PARI) a(n)=(n-1)<<(n+2)+n+5 \\ _Charles R Greathouse IV_, Nov 21 2011
%Y A045618 Cf. A000337.
%K A045618 easy,nonn
%O A045618 0,2
%A A045618 _Wolfdieter Lang_