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A047974 a(n) = a(n-1) + 2*(n-1)*a(n-2). 23
1, 1, 3, 7, 25, 81, 331, 1303, 5937, 26785, 133651, 669351, 3609673, 19674097, 113525595, 664400311, 4070168161, 25330978113, 163716695587, 1075631907655, 7296866339961, 50322142646161, 356790528924523, 2570964805355607, 18983329135883665, 142389639792952801, 1091556096587136051 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

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

Hankel transform is A108400. - Paul Barry, Feb 11 2008

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

Equals inverse binomial transform of A000898. - Gary W. Adamson, Oct 06 2008

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 -infinity..infinity - Groux Roland, Mar 26 2011

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

From Peter Bala, Dec 03 2017: (Start)

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.

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)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

T. Amdeberhan, V. de Angelis, A. Dixit, V. H. Moll and C. Vignat, From sequences to polynomials and back, via operator orderings, J. Math. Phys. 54, 123502 (2013); Alternative copy

P. Bala, Integer sequences that become periodic on reduction modulo k for all k

Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts); Alternative copy

Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, Discrete Applied Mathematics, Volume 161, Issues 10-11, July 2013, Pages 1378-1394; Alternative copy.

Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.

Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8.

T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150 [math.RT], 2013.

A. Khruzin, Enumeration of chord diagrams, arXiv:math/0008209 [math.CO], 2000.

R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.

J. Quaintance, H. Kwong, Permutations and combinations of colored mulisets, JIS 13 (2010) #10.2.6.

Index entries for related partition-counting sequences

Index entries for sequences related to Hermite polynomials

FORMULA

E.g.f.: exp(x^2+x). - Len Smiley, Dec 11 2001

Binomial transform of A001813 (with interpolated zeros). - Paul Barry, May 09 2003

a(n) = Sum_{k=0..n} C(k,n-k)*n!/k!. - Paul Barry, Mar 29 2007

a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*(2k)!/k!; - Paul Barry, Feb 11 2008

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

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

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

a(n) ~ 2^(n/2-1/2)*exp(sqrt(n/2)-n/2-1/8)*n^(n/2). - Vaclav Kotesovec, Oct 08 2012

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

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

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

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

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

MAPLE

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

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

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

seq(A047974(n), n=0..26); # Peter Luschny, Nov 29 2017

MATHEMATICA

Range[0, 23]!*CoefficientList[ Series[ Exp[x*(1-x^2)/(1 - x)], {x, 0, 23 }], x] - (* Zerinvary Lajos, Mar 23 2007 *)

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

PROG

(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

(PARI) x='x+O('x^66); Vec(serlaplace(exp(x^2+x))) \\ Joerg Arndt, May 04 2013

CROSSREFS

Row sums of A067147.

Cf. A000680, A000898, A001147, A115329, A132101.

Sequence in context: A148734 A124425 A118398 * A148735 A148736 A148737

Adjacent sequences:  A047971 A047972 A047973 * A047975 A047976 A047977

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 12 21:10 EST 2018. Contains 317116 sequences. (Running on oeis4.)