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A052841
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E.g.f.: 1/(exp(x)*(2-exp(x))).
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51
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1, 0, 2, 6, 38, 270, 2342, 23646, 272918, 3543630, 51123782, 811316286, 14045783798, 263429174190, 5320671485222, 115141595488926, 2657827340990678, 65185383514567950, 1692767331628422662, 46400793659664205566, 1338843898122192101558, 40562412499252036940910
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OFFSET
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0,3
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COMMENTS
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Stirling transform of A005359(n)=[0,2,0,24,0,720,...] is a(n)=[0,2,6,38,270,...].
Stirling transform of -(-1)^n*A052657(n-1)=[0,0,2,-6,48,-240,...] is a(n-1)=[0,0,2,6,38,270,...].
Stirling transform of -(-1)^n*A052558(n-1)=[1,-1,4,-12,72,-360,...] is a(n-1)=[1,0,2,6,38,270,...].
Stirling transform of 2*A052591(n)=[2,4,24,96,...] is a(n+1)=[2,6,38,270,...].
(End)
Also the central moments of a Geometric(1/2) random variable (for example the number of coin tosses until the first head). - Svante Janson, Dec 10 2012
Also the number of ordered set partitions of {1..n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). - Gus Wiseman, Feb 13 2019
Also the number of ordered set partitions of {1..n} with an even number of blocks. - Geoffrey Critzer, Jul 04 2020
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LINKS
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FORMULA
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O.g.f.: Sum_{n>=0} (2*n)! * x^(2*n) / Product_{k=1..2*n} (1-k*x). - Paul D. Hanna, Jul 20 2011
Also, a(n) = Sum_{k=0..[n/2]} (2k)!*Stirling2(n, 2k). - Ralf Stephan, May 23 2004
a(n) = D^n*(1/(1-x^2)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000670 and A005649. - Peter Bala, Nov 25 2011
E.g.f.: 1/(2*G(0)), where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k*(k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
a(n) = (h(n)+(-1)^n)/2 where h(n) = Sum_{k=0..n} E(n,k)*2^k and E(n,k) the Eulerian numbers A173018 (see also A156365). - Peter Luschny, Sep 19 2015
a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 11 2020
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EXAMPLE
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The a(4) = 38 ordered set partitions with no cyclical adjacencies:
{{1}{2}{3}{4}} {{1}{24}{3}} {{13}{24}}
{{1}{2}{4}{3}} {{1}{3}{24}} {{24}{13}}
{{1}{3}{2}{4}} {{13}{2}{4}}
{{1}{3}{4}{2}} {{13}{4}{2}}
{{1}{4}{2}{3}} {{2}{13}{4}}
{{1}{4}{3}{2}} {{2}{4}{13}}
{{2}{1}{3}{4}} {{24}{1}{3}}
{{2}{1}{4}{3}} {{24}{3}{1}}
{{2}{3}{1}{4}} {{3}{1}{24}}
{{2}{3}{4}{1}} {{3}{24}{1}}
{{2}{4}{1}{3}} {{4}{13}{2}}
{{2}{4}{3}{1}} {{4}{2}{13}}
{{3}{1}{2}{4}}
{{3}{1}{4}{2}}
{{3}{2}{1}{4}}
{{3}{2}{4}{1}}
{{3}{4}{1}{2}}
{{3}{4}{2}{1}}
{{4}{1}{2}{3}}
{{4}{1}{3}{2}}
{{4}{2}{1}{3}}
{{4}{2}{3}{1}}
{{4}{3}{1}{2}}
{{4}{3}{2}{1}}
(End)
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MAPLE
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spec := [S, {B=Prod(C, C), C=Set(Z, 1 <= card), S=Sequence(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
P := proc(n, x) option remember; if n = 0 then 1 else
(n*x+2*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x); expand(%) fi end:
A052841 := n -> subs(x=2, P(n, x)):
h := n -> add(combinat:-eulerian1(n, k)*2^k, k=0..n):
a := n -> (h(n)+(-1)^n)/2: seq(a(n), n=0..21); # Peter Luschny, Sep 19 2015
b := proc(n, m) option remember; if n = 0 then 1 else
(m - 1)*b(n - 1, m) + (m + 1)*b(n - 1, m + 1) fi end:
a := n -> b(n, 0): seq(a(n), n = 0..21); # Peter Luschny, Jun 23 2023
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MATHEMATICA
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a[n_] := If[n == 0, 1, (PolyLog[-n, 1/2]/2 + (-1)^n)/2]; (* or *)
With[{nn=30}, CoefficientList[Series[1/(Exp[x](2-Exp[x])), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Apr 08 2019 *)
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PROG
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(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst(1/(1-y^2), y, exp(x+x*O(x^n))-1), n))
(PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!*x^(2*m)/prod(k=1, 2*m, 1-k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */
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CROSSREFS
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Inverse binomial transform of A000670.
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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