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 A115329 #59 Jul 12 2024 21:59:29
%S A115329 1,1,5,13,73,281,1741,8485,57233,328753,2389141,15539261,120661465,
%T A115329 866545993,7140942173,55667517781,484124048161,4046845186145,
%U A115329 36967280461093,328340133863533,3137853448906601,29405064157989241
%N A115329 Expansion of e.g.f.: exp(x + 2*x^2).
%C A115329 Term-by-term square of sequence with e.g.f.: exp(x+m/2*x^2) is given by e.g.f.: exp(x/(1-m*x))/sqrt(1-m^2*x^2) for all m.
%C A115329 Combinatorial interpretation: a(n) counts the partitions of a set of n distinguishable objects into subsets of size 1 and 2 with the additional feature that the constituents of the subset of size 2 acquire 2 colors. - _Karol A. Penson_ and P. Blasiak (blasiak(AT)lptl.jussieu.fr), Jun 03 2006
%C A115329 In general, e.g.f. exp(x+m*x^2) has general term sum{k=0..n, C(n,k)*m^k*(n-k)!/(n-m*k)!}. [_Paul Barry_, Nov 07 2008]
%C A115329 The sequence terms have the form 4*m + 1 (follows from the recurrence). a(n+k) = a(n) (mod k) holds for all n and k by an induction argument making use of the recurrence equation. For each k the sequence a(n) taken modulo k is thus periodic with exact period dividing k. - _Peter Bala_, Nov 15 2017
%H A115329 Seiichi Manyama, <a href="/A115329/b115329.txt">Table of n, a(n) for n = 0..665</a> (terms 0..200 from Vincenzo Librandi)
%H A115329 Magdalena Boos, Giovanni Cerulli Irelli, Francesco Esposito, <a href="https://arxiv.org/abs/1802.06425">Parabolic orbits of 2-nilpotent elements for classical groups</a>, arXiv:1802.06425 [math.RT], 2018.
%F A115329 Term-by-term square equals A115330 which has e.g.f.: exp(x/(1-4*x))/sqrt(1-16*x^2).
%F A115329 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)2^k*n!/(n-k)! = Sum_{k=0..n} C(n,k)2^k*(n-k)!/(n-2k)!. - _Paul Barry_, Nov 07 2008
%F A115329 a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+8*x)*d/dx. Cf. A000085 and A047974. - _Peter Bala_, Dec 07 2011
%F A115329 a(n) = a(n-1) + 4*(n-1)*a(n-2). - _R. J. Mathar_, Dec 10 2011
%F A115329 a(n) ~ 2^(n-1/2)*exp(sqrt(n)/2-n/2-1/16)*n^(n/2). - _Vaclav Kotesovec_, Oct 19 2012
%F A115329 G.f.: 1/Q(0), where Q(k)= 1 + 4*x*k - x/(1 - 4*x*(k+1)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 17 2013
%F A115329 G.f.: 1/G(0), where G(k)= 1 - x - 4*(k+1)*x^2/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 21 2013
%F A115329 a(n) = i^(1 - n)*2^(3*(n - 1)/2)*KummerU((1 - n)/2, 3/2, -1/8). - _Peter Luschny_, Nov 21 2017
%F A115329 a(n) = (-i*sqrt(2))^n * Hermite(n, i/(2*sqrt(2))). - _G. C. Greubel_, Jul 12 2024
%p A115329 a := n -> I^(1 - n)*2^((3*(n - 1))/2)*KummerU((1 - n)/2, 3/2, -1/8):
%p A115329 seq(simplify(a(n)), n=0..21); # _Peter Luschny_, Nov 21 2017
%t A115329 Range[0, 20]! CoefficientList[Series[Exp[(x + 2 x^2)], {x, 0, 20}], x] (* _Vincenzo Librandi_, May 22 2013 *)
%o A115329 (PARI) a(n)=local(m=4);n!*polcoeff(exp(x+m/2*x^2+x*O(x^n)),n)
%o A115329 (Magma)
%o A115329 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A115329 Coefficients(R!(Laplace( Exp(x+2*x^2) ))); // _G. C. Greubel_, Jul 12 2024
%o A115329 (SageMath)
%o A115329 [(-i*sqrt(2))^n*hermite(n, i/(2*sqrt(2))) for n in range(31)] # _G. C. Greubel_, Jul 12 2024
%Y A115329 Column k=4 of A359762.
%Y A115329 Cf. A000085, A047974, A115330.
%Y A115329 Sequences with e.g.f = exp(x + q*x^2): A158968 (q=-9), A158954 (q=-4), A362177 (q=-3), A362176 (q=-2), A293604 (q=-1), A000012 (q=0), A047974 (q=1), this sequence (q=2), A293720 (q=4).
%K A115329 nonn,easy
%O A115329 0,3
%A A115329 _Paul D. Hanna_, Jan 20 2006
%E A115329 More terms from _Karol A. Penson_ and P. Blasiak (blasiak(AT)lptl.jussieu.fr), Jun 03 2006