A047974 - OEIS

%I #165 Jul 12 2024 21:58:47

%S 1,1,3,7,25,81,331,1303,5937,26785,133651,669351,3609673,19674097,

%T 113525595,664400311,4070168161,25330978113,163716695587,

%U 1075631907655,7296866339961,50322142646161,356790528924523,2570964805355607,18983329135883665,142389639792952801,1091556096587136051

%N a(n) = a(n-1) + 2*(n-1)*a(n-2).

%C Related to partially ordered sets. - Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003

%C The number of partial permutation matrices P in GL_n with P^2=0. Alternatively, the number of orbits of the Borel group of upper triangular matrices acting by conjugation on the set of matrices M in GL_n with M^2=0. - Brian Rothbach (rothbach(AT)math.berkeley.edu), Apr 16 2004

%C Number of ways to use the elements of {1..n} once each to form a collection of sequences, each having length 1 or 2. - Bob Proctor, Apr 18 2005

%C Hankel transform is A108400. - _Paul Barry_, Feb 11 2008

%C This is also the number of subsets of equivalent ways to arrange the elements of n pairs, when equivalence is defined under the joint operation of (optional) reversal of elements combined with permutation of the labels and the subset maps to itself. - _Ross Drewe_, Mar 16 2008

%C Equals inverse binomial transform of A000898. - _Gary W. Adamson_, Oct 06 2008

%C a(n) is also the moment of order n for the measure of density exp(-(x-1)^2/4)/(2*sqrt(Pi)) over the interval -oo..oo. - _Groux Roland_, Mar 26 2011

%C The n-th term gives the number of fixed-point-free involutions in S_n^B, the group of permutations on the set {-n,...,-1,1,2,...,n}. - _Matt Watson_, Jul 26 2012

%C From _Peter Bala_, Dec 03 2017: (Start)

%C a(n+k) == a(n) (mod k) for all n and k. Hence for each k, the sequence a(n) taken modulo k is a periodic sequence and the exact period divides k. Cf. A115329.

%C More generally, the same divisibility property holds for any sequence with an e.g.f. of the form F(x)*exp(x*G(x)), where F(x) and G(x) are power series with integer coefficients and G(0) = 1. See the Bala link for a proof. (End)

%H Vincenzo Librandi, <a href="/A047974/b047974.txt">Table of n, a(n) for n = 0..200</a>

%H T. Amdeberhan, V. de Angelis, A. Dixit, V. H. Moll and C. Vignat, <a href="http://dx.doi.org/10.1063/1.4836778">From sequences to polynomials and back, via operator orderings</a>, J. Math. Phys. 54, 123502 (2013); <a href="http://www.tulane.edu/~vhm/papers_html/ordering1.pdf">Alternative copy</a>

%H Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>

%H Jonathan Burns, <a href="http://knot.math.usf.edu/data/SimpleAssemblyTable.txt">Assembly Graph Words - Single Transverse Component (Counts)</a>; <a href="http://shell.cas.usf.edu/~saito/DNAweb/SimpleAssemblyTable.txt">Alternative copy</a>

%H Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, <a href="http://dx.doi.org/10.1016/j.dam.2013.01.003">Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination</a>, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394; <a href="http://jtburns.myweb.usf.edu/assembly/papers/Graphs_and_DNA_Recomb_2011.pdf">Alternative copy</a>.

%H Jonathan Burns and Tilahun Muche, <a href="http://arxiv.org/abs/1105.2926">Counting Irreducible Double Occurrence Words</a>, arXiv preprint arXiv:1105.2926 [math.CO], 2011.

%H Samuele Giraudo, <a href="https://arxiv.org/abs/1709.08416">Combalgebraic structures on decorated cliques</a>, arXiv:1709.08416 [math.CO], 2017; and <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2017/15%20Giraudo.html">also</a>, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8.

%H T. Halverson and M. Reeks, <a href="http://arxiv.org/abs/1302.6150">Gelfand Models for Diagram Algebras</a>, arXiv preprint arXiv:1302.6150 [math.RT], 2013.

%H A. Khruzin, <a href="https://arxiv.org/abs/math/0008209">Enumeration of chord diagrams</a>, arXiv:math/0008209 [math.CO], 2000.

%H G. Latouche and P. G. Taylor, <a href="https://doi.org/10.1007/s11134-009-9153-6">A stochastic fluid model for an ad hoc mobile network</a>, Queueing Syst. 63, No. 1-4, 109-129 (2009), eq. (1).

%H R. A. Proctor, <a href="https://arxiv.org/abs/math/0606404">Let's Expand Rota's Twelvefold Way for Counting Partitions!</a>, arXiv:math/0606404 [math.CO], 2006-2007.

%H J. Quaintance and H. Kwong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Kwong/kwong6.html">Permutations and combinations of colored multisets</a>, JIS 13 (2010) #10.2.6.

%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>

%F E.g.f.: exp(x^2+x). - _Len Smiley_, Dec 11 2001

%F Binomial transform of A001813 (with interpolated zeros). - _Paul Barry_, May 09 2003

%F a(n) = Sum_{k=0..n} C(k,n-k)*n!/k!. - _Paul Barry_, Mar 29 2007

%F a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*(2k)!/k!; - _Paul Barry_, Feb 11 2008

%F G.f.: 1/(1-x-2*x^2/(1-x-4*x^2/(1-x-6*x^2/(1-x-8*x^2/(1-... (continued fraction). -_Paul Barry_, Apr 10 2009

%F E.g.f.: Q(0); Q(k) = 1+(x^2+x)/(2*k+1-(x^2+x)*(2*k+1)/((x^2+x)+(2*k+2)/Q(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 24 2011

%F a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+4*x)*d/dx. Cf. A000085 and A115329. - _Peter Bala_, Dec 07 2011

%F a(n) ~ 2^(n/2 - 1/2)*exp(sqrt(n/2) - n/2 - 1/8)*n^(n/2). - _Vaclav Kotesovec_, Oct 08 2012

%F E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + (1+x)/(k+1)/(1-x/(x+1/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 26 2013

%F a(n) = i^(-n)*H_{n}(i/2) with i the imaginary unit and H_{n} the Hermite polynomial of degree n. - _Alyssa Byrnes_ and C. Vignat, Jan 31 2013

%F E.g.f.: -Q(0)/x where Q(k) = 1 - (1+x)/(1 - x/(x - (k+1)/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Mar 06 2013

%F G.f.: 1/Q(0), where Q(k) = 1 + x*2*k - x/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 17 2013

%F E.g.f.: E(0)-1-x-x^2, where E(k) = 2 + 2*x*(1+x) - 8*k^2 + x^2*(1+x)^2*(2*k+3)*(2*k-1)/E(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 21 2013

%F E.g.f.: Product_{k>=1} 1/(1 + (-x)^k)^(mu(k)/k). - _Ilya Gutkovskiy_, May 26 2019

%F a(n) = Sum_{k=0..floor(n/2)} 2^k*B(n, k), where B are the Bessel numbers A100861. - _Peter Luschny_, Jun 04 2021

%p seq( add(n!/((n-2*k)!*k!), k=0..floor(n/2)), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001

%p with(combstruct):seq(count(([S,{S=Set(Union(Z,Prod(Z,Z)))},labeled],size=n)),n=0..30); # Detlef Pauly (dettodet(AT)yahoo.de), Sep 25 2003

%p A047974 := n -> I^(-n)*orthopoly[H](n, I/2):

%p seq(A047974(n), n=0..26); # _Peter Luschny_, Nov 29 2017

%t Range[0, 23]!*CoefficientList[ Series[ Exp[x*(1-x^2)/(1 - x)], {x, 0,23 }], x] - (* _Zerinvary Lajos_, Mar 23 2007 *)

%t Table[I^(-n)*HermiteH[n, I/2], {n, 0, 23}] - (* _Alyssa Byrnes_ and C. Vignat, Jan 31 2013 *)

%o (MATLAB) N = 18; A = zeros(N,1); for n = 1:N; a = factorial(n); s = 0; k = 0; while k <= floor(n/2); b = factorial(n - 2*k); c = factorial(k); s = s + a/(b*c); k = k+1; end; A(n) = s; end; disp(A); % _Ross Drewe_, Mar 16 2008

%o (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(x^2+x))) \\ _Joerg Arndt_, May 04 2013

%o (Magma) [n le 2 select 1 else Self(n-1) + 2*(n-2)*Self(n-2): n in [1..40]]; // _G. C. Greubel_, Jul 12 2024

%o (SageMath) [(-i)^n*hermite(n,i/2) for n in range(41)] # _G. C. Greubel_, Jul 12 2024

%Y Row sums of A067147.

%Y Column k=2 of A359762.

%Y Cf. A000085, A000680, A000898, A001147, A001813, A100861, A108400, A115329, A132101.

%Y 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), this sequence (q=1), A115329 (q=2), A293720 (q=4).

%K nonn

%O 0,3

%A _N. J. A. Sloane_