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A047974
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a(n) = a(n-1)+2*(n-1)*a(n-2).
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12
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1, 1, 3, 7, 25, 81, 331, 1303, 5937, 26785, 133651, 669351, 3609673, 19674097, 113525595, 664400311, 4070168161, 25330978113, 163716695587, 1075631907655, 7296866339961, 50322142646161, 356790528924523, 2570964805355607
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 (pbarry(AT)wit.ie), 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 (rd(AT)labyrinth.net.au), Mar 16 2008
Equals inverse binomial transform of A000898 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 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.
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REFERENCES
| Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts), http://shell.cas.usf.edu/~saito/DNAweb/SimpleAssemblyTable.txt.
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, May 23, 2011; http://jtburns.myweb.usf.edu/assembly/papers/Graphs_and_DNA_Recomb_2011.pdf.
Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, Arxiv preprint arXiv:1105.2926, 2011.
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LINKS
| A. Khruzin, Enumeration of chord diagrams
Index entries for related partition-counting sequences
Index entries for sequences related to Hermite polynomials
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FORMULA
| seq( sum('n!/((n-2*k)!*k!)', 'k'=0..floor(n/2)), n=0..30 ); - Detlef Pauly (dettodet(AT)yahoo.de), Nov 15 2001
E.g.f.: exp(x^2+x). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 11 2001
Binomial transform of A001813 (with interpolated zeros). - Paul Barry (pbarry(AT)wit.ie), May 09 2003
a(n)=sum{k=0..n, C(k,n-k)*n!/k!}; - Paul Barry (pbarry(AT)wit.ie), Mar 29 2007
a(n)=sum{k=0..floor(n/2), C(n,2k)(2k)!/k!}; - Paul Barry (pbarry(AT)wit.ie), 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). [From Paul Barry (pbarry(AT)wit.ie), Apr 10 2009]
E.g.f.: exp((x^2)+x)=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
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MAPLE
| 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
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MATHEMATICA
| Range[0, 23]!*CoefficientList[ Series[ Exp[x*(1-x^2)/(1 - x)], {x, 0, 23 }], x] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 23 2007
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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 (rd(AT)labyrinth.net.au), Mar 16 2008
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CROSSREFS
| Row sums of A067147.
Cf. A000680, A001147, A132101.
A000898 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 06 2008]
Sequence in context: A148734 A124425 A118398 * A148735 A148736 A148737
Adjacent sequences: A047971 A047972 A047973 * A047975 A047976 A047977
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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