A002832 -id:A002832 - cp's OEIS frontend

A000940 Number of n-gons with n vertices.

1, 2, 4, 12, 39, 202, 1219, 9468, 83435, 836017, 9223092, 111255228, 1453132944, 20433309147, 307690667072, 4940118795869, 84241805734539, 1520564059349452, 28963120073957838, 580578894859915650, 12217399235411398127, 269291841184184374868, 6204484017822892034404

Offset: 3

N. J. A. Sloane

Number of inequivalent undirected Hamiltonian cycles in complete graph on n labeled nodes under action of dihedral group of order 2n acting on nodes.

			Label the vertices of a regular n-gon 1,2,...,n.
For n=3,4,5 representatives for the polygons counted here are:
  (1,2,3,1),
  (1,2,3,4,1), (1,2,4,3,1),
  (1,2,3,4,5,1), (1,2,3,5,4,1), (1,2,4,5,3,1), (1,3,5,2,4,1).
For n=6:
  (1,2,3,4,5,6,1), (1,2,3,4,6,5,1), (1,2,3,5,6,4,1),
  (1,2,3,6,5,4,1), (1,2,4,3,6,5,1), (1,2,4,6,3,5,1),
  (1,2,4,6,5,3,1), (1,2,5,3,6,4,1), (1,2,5,4,6,3,1),
  (1,2,5,6,3,4,1), (1,2,6,4,5,3,1), (1,3,5,2,6,4,1).

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ludovic Schwob, Table of n, a(n) for n = 3..500 (terms 3..100 from T. D. Noe).
S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353.
S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353. [Annotated scanned copy]
Samuel Herman and Eirini Poimenidou, Orbits of Hamiltonian Paths and Cycles in Complete Graphs, arXiv:1905.04785 [math.CO], 2019.
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence, except he has a(6)=7 instead of 12)
R. C. Read, Letter to N. J. A. Sloane, 1992
R. C. Read, Combinatorial problems in theory of music, Discrete Math. 167 (1997), 543-551.
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 9.
N. J. A. Sloane, Illustration of initial terms [Annotated page from Golomb-Welch article]
Venta Terauds and J. Sumner, Circular genome rearrangement models: applying representation theory to evolutionary distance calculations, arXiv preprint arXiv:1712.00858 [q-bio.PE], 2017.
Venta Terauds and Jeremy Sumner, A new algebraic approach to genome rearrangement models, arXiv:2012.11665 [q-bio.PE], 2020.

Cf. A000939, A007619. Bisections give A094156, A094157.

For permutation classes under various symmetries see A089066, A262480, A002619.

with(numtheory);
# for n odd:
Sd:=proc(n) local t1,d; t1:=2^((n-1)/2)*n^2*((n-1)/2)!; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(4*n^2); end;
# for n even:
Se:=proc(n) local t1,d; t1:=2^(n/2)*n*(n+6)*(n/2)!/4; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(4*n^2); end;
A000940:=n-> if n mod 2 = 0 then Se(n) else Sd(n); fi;

a[n_] := (t1 = If[OddQ[n], 2^((n - 1)/2)*n^2*((n - 1)/2)!, 2^(n/2)*n*(n + 6)*(n/2)!/4]; For[ d = 1 , d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[n/d]^2*d!*(n/d)^d]]; t1/(4*n^2)); Table[a[n], {n, 3, 25}] (* Jean-François Alcover, Jun 19 2012, after Maple *)

a(n)={if(n<3, 0, (2^(n\2-2)*(n\2)!*n*if(n%2, 4*n, n + 6) + sumdiv(n, d, eulerphi(n/d)^2*d!*(n/d)^d))/(4*n^2))} \\ Andrew Howroyd, Sep 09 2018

from sympy import factorial, divisors, totient
def A000940(n): return 1 if n == 3 else ((sum(totient(m:=n//d)**2*factorial(d)*m**d for d in divisors(n,generator=True))+(1<<(k:=n>>1)-2)*n*(n<<2 if n&1 else (n+6))*factorial(k))>>2)//n//n # Chai Wah Wu, Nov 07 2022

For formula see Maple lines.
a(p) = ((((p-1)! + 1)/p) + p - 2 + (2^((p-1)/2)*((p-1)/2)!))/4 for prime p. See A007619. - Ian Mooney, Oct 05 2022
a(n) ~ sqrt(2*Pi)/4 * n^(n-3/2) / e^n. - Ludovic Schwob, Nov 03 2022

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 05 2004

A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

Original entry on oeis.org

1, 1, 4, 46, 1024, 36976, 1965664, 144361456, 13997185024, 1731678144256, 266182076161024, 49763143319190016, 11118629668610842624, 2925890822304510631936, 895658946905031792553984, 315558279782214450517374976, 126780706777739389745128013824

Offset: 0

N. J. A. Sloane, Don Knuth

Also central terms of the triangle in A008280. - Reinhard Zumkeller, Nov 01 2013

Conjecture: taking the sequence modulo an integer k gives an eventually purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 9 begins [1, 1, 4, 1, 7, 4, 1, 7, 4, 1, 7, ...] with an apparent period [4, 1, 7] of length 3 = phi(9)/2 beginning at a(2). - Peter Bala, May 08 2023

V. I. Arnold, Springer numbers and Morsification spaces. J. Algebraic Geom. 1 (1992), no. 2, 197-214.
L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Cf. A084938, A002832. For a signed version see A099023.
Related polynomials in A098277.
A diagonal of A323834.
Cf. A005799.

a000657 n = a008280 (2 * n) n  -- Reinhard Zumkeller, Nov 01 2013

Digits := 40: rr := array(1..40,1..40): rr[1,1] := 1: for i from 1 to 39 do rr[i+1,1] := subs(x=0,diff(1+tan(x),x$i)): od: for i from 2 to 40 do for j from 2 to i do rr[i,j] := rr[i,j-1]-(-1)^i*rr[i-1,j-1]: od: od: [seq(rr[2*i-1,i],i=1..20)];
# Alternatively after Alois P. Heinz in A000111:
b := proc(u, o) option remember;
`if`(u + o = 0, 1, add(b(o - 1 + j, u - j), j = 1..u)) end:
a := n -> b(n, n): seq(a(n), n = 0..15); # Peter Luschny, Oct 27 2017

max = 20; rr[1, 1] = 1; For[i = 1, i <= 2*max - 1, i++, rr[i + 1, 1] = D[1 + Tan[x], {x, i}] /. x -> 0]; For[i = 2, i <= 2*max, i++, For[j = 2, j <= i, j++, rr[i, j] = rr[i, j - 1] - (-1)^i*rr[i - 1, j - 1]]]; Table[rr[2*i - 1, i], {i, 1, max}] (* Jean-François Alcover, Jul 10 2012, after Maple *)
T[n_,0] := KroneckerDelta[n,0]; T[n_,k_] := T[n,k]=T[n,k-1]+T[n-1,n-k]; Table[T[2n,n], {n,0,16}] (* Oliver Seipel, Nov 24 2024, after Peter Luschny *)

a(n):=(-1)^(n)*sum(binomial(n,k)*euler(n+k),k,0,n); /* Vladimir Kruchinin, Apr 06 2015 */

# Algorithm of L. Seidel (1877)
def A000657_list(n) :
    R = []; A = {-1:0, 0:1}
    k = 0; e = 1
    for i in (0..n) :
        Am = 0; A[k + e] = 0; e = -e
        for j in (0..i) :
            Am += A[k]; A[k] = Am; k += e
        if e < 0 :
            R.append(A[0])
    return R
A000657_list(30)  # Peter Luschny, Apr 02 2012

Row sums of triangle, read by rows, [0, 1, 4, 9, 16, 25, 36, 49, ...] DELTA [1, 2, 6, 5, 11, 8, 16, 11, 21, 14, ...] where DELTA is Deléham's operator defined in A084938.
G.f.: Sum_{n>=0} a(n)*x^n = 1/(1-1*1x/(1-1*3x/(1-2*5x/(1-2*7x/(1-3*9x/...))))). - Ralf Stephan, Sep 09 2004
G.f.: 1/G(0) where G(k) = 1 - x*(8*k^2+4*k+1) - x^2*(k+1)^2*(4*k+1)*(4*k+3)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013
G.f.: G(0)/(1-x), where G(k) = 1 - x^2*(k+1)^2*(4*k+1)*(4*k+3)/( x^2*(k+1)^2*(4*k+1)*(4*k+3) - (1 - x*(8*k^2+4*k+1))*(1 - x*(8*k^2+20*k+13))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Feb 01 2014
a(n) = (-1)^(n)*Sum_{k=0..n} C(n,k)*Euler(n+k). - Vladimir Kruchinin, Apr 06 2015
a(n) ~ 2^(4*n+5/2) * n^(2*n+1/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Apr 06 2015
Conjectural e.g.f. as a continued fraction: 1/(1 - (1 - exp(-2*t))/(2 - (1 - exp(-4*t))/(1 - (1 - exp(-6*t))/(2 - (1 - exp(-8*t))/(1 - ... )))) = 1 + t + 4*t^2/2! + 46*t^3/3! + .... Cf. A005799. - Peter Bala, Dec 26 2019

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 12 2001

Corrected by Sean A. Irvine, Dec 22 2010

A163747 Expansion of e.g.f. 2exp(x)(1-exp(x))/(1+exp(2*x)).

Original entry on oeis.org

0, -1, -1, 2, 5, -16, -61, 272, 1385, -7936, -50521, 353792, 2702765, -22368256, -199360981, 1903757312, 19391512145, -209865342976, -2404879675441, 29088885112832, 370371188237525, -4951498053124096, -69348874393137901

Offset: 0

Roger L. Bagula, Aug 03 2009

sign

The real part of the exponential expansion of 2*((1+i)/(1+i*exp(z))-1) = (-1-i)*z + (-1/2+i/2)*z^2 + (1/3+i/3)*z^3 + (5/24-5i/24)*z^4 + (-2/15-2i/15)*z^5 + ...  where i is the imaginary unit.
From Paul Curtz, Mar 12 2013: (Start)
a(n) is an autosequence of the first kind; a(n) and successive differences are:
0,   -1,   -1,      2,      5,    -16,    -61;
-1,   0,    3,      3,    -21,    -45,    333;
1,    3,    0,    -24,    -24,    378,    780;
2,   -3,  -24,      0,    402,    402, -11214;
-5, -21,   24,    402,      0, -11616, -11616;
-16, 45,  378,   -402, -11616,      0, 514608;
61, 333, -780, -11214,  11616, 514608,      0;
The main diagonal is A000004. The inverse binomial transform is the signed sequence.
The first two upper diagonals are A002832 (median Euler numbers) signed.
Sum of the antidiagonals: 0, -2, 0, 10, 0, ... = 2*A122045(n+1) (End)

Robert Israel, Table of n, a(n) for n = 0..485
Toufik Mansour, Howard Skogman, and Rebecca Smith, Passing through a stack k times with reversals, arXiv:1808.04199 [math.CO], 2018.
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (negative of the zeroth column of matrix a_{n,k} on p. 18).

Variant: A163982.
Cf. A000004, A000111, A002832, A122045.
Minus the zeroth column of A323833.

A163747 := proc(n) exp(t)*(1-exp(t))/(1+exp(2*t)) ; coeftayl(%,t=0,n) ; 2*%*n! ; end proc: # R. J. Mathar, Sep 11 2011
seq((euler(n) - 2^n*(2*euler(n,1)+euler(n,3/2)))/2 + 1, n=0..30); # Robert Israel, May 24 2016
egf := (2 - 2*I)/(exp(-x) + I); ser := series(egf, x, 24):
seq(n!*Re(coeff(ser, x, n)), n = 0..22); # Peter Luschny, Aug 09 2021

f[t_] = (1 + I)/(1 + I*Exp[t]) - 1;
Table[Re[2*n!*SeriesCoefficient[Series[f[t], {t, 0, 30}], n]], {n, 0, 30}]
max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 - x + (4*k+3)*(k+1)*x^2 /( 1 + (4*k+5)*(k+1)*x^2 / g[k+1]); gf = -x/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jan 22 2015, after Sergei N. Gladkovskii *)
Table[(EulerE[n] - 2^n (2 EulerE[n, 1] + EulerE[n, 3/2]))/2 + 1, {n, 0, 20}] (* Benedict W. J. Irwin, May 24 2016 *)

G.f.: -x/W(0), where W(k) = 1 - x + (4*k+3)*(k+1)*x^2 / (1 + (4*k+5)*(k+1)*x^2 / W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 22 2015
a(n) ~ n! * (cos(Pi*n/2) - sin(Pi*n/2)) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Apr 23 2015
a(n) = (A122045(n) - 2^n(2*Euler(n,1) + Euler(n,3/2)))/2 + 1, where Euler(n,x) is the n-th Euler polynomial. - Benedict W. J. Irwin, May 24 2016
a(n) = 2*4^n*(HurwitzZeta(-n, 1/4) - HurwitzZeta(-n, 3/4)) + HurwitzZeta(-n, 1)*(4^(n+1) - 2^(n+1)). - Peter Luschny, Jul 21 2020
a(n) = 2^n*(Euler(n, 1/2) - Euler(n, 1)). - Peter Luschny, Mar 19 2021
a(n) = ((-2)^(n + 1)*(1 - 2^(n + 1))*Bernoulli(n + 1))/(n + 1) + Euler(n). - Peter Luschny, May 06 2021
a(n) = n!*Re([x^n]((2 - 2*i)/(i + exp(-x)))). - Peter Luschny, Aug 09 2021

A098277 Coefficients of polynomials D(n,x) related to median Euler numbers.

Original entry on oeis.org

1, 2, 2, 8, 20, 12, 48, 224, 344, 168, 384, 2880, 8096, 9872, 4272, 3840, 42240, 186816, 407936, 430688, 171168, 46080, 698880, 4451328, 15030528, 27944576, 26627648, 9915072, 645120, 12902400, 111605760, 535271424, 1519126272

Offset: 0

Ralf Stephan, Sep 07 2004

2^n(x+1) divides D(n,x).

			D(0,x) = 1,
D(1,x) = 2*x + 2,
D(2,x) = 8*x^2 + 20*x + 12,
D(3,x) = 48*x^3 + 224*x^2 + 344*x + 168,
D(4,x) = 384*x^4 + 2880*x^3 + 8096*x^2 + 9872*x + 4272.

A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26.

D(n, 1/2) = A002832(n+1), D(n, -1/2) = A000657(n).
D(n, 0)/2^n = A098278(n), D(n, 1)/2^n = A098279(n).
Leading coefficients are A000165. Constant terms are in A098431.

d[0, ] = 1; d[n, x_] := d[n, x] = (x+1)(x+2)d[n-1, x+2] - x(x+1)d[n-1, x];
Table[CoefficientList[d[n, x], x] // Reverse, {n, 0, 8}] // Flatten (* Jean-François Alcover, Jul 27 2018 *)

D(n,x)=if(n<1,1,(x+1)*(x+2)*D(n-1,x+2)-x*(x+1)*D(n-1,x))

T(n,k)=local(A=sum(m=0,n,m!*(2*x)^m*prod(j=1,m,(j+y)/(1+j*(j+1)*x +x*O(x^n)))));polcoeff(polcoeff(A,n,x),n-k,y)
{for(n=0,8,for(k=0,n,print1(T(n,k),", "));print())} \\ Paul D. Hanna, Sep 05 2012

Recurrence: D(0, x)=1, D(n, x) = (x+1)(x+2)D(n-1, x+2) - x(x+1)D(n-1, x).
G.f.: Sum[n>=0, D(n, x)t^n] = 1/(1-2(x+1)t/(1-2(x+2)t/(1-4(x+3)t/(1-4(x+4)t/...)))).
G.f.: Sum_{n>=0} D(n,y)*x^n = Sum_{n>=0} n!*(2*x)^n*Product_{k=1..n} (k+y)/(1+k*(k+1)*x). - Paul D. Hanna, Sep 05 2012

A323833 A Seidel matrix A(n,k) read by antidiagonals upwards.

Original entry on oeis.org

0, 1, 1, 1, 0, -1, -2, -3, -3, -2, -5, -3, 0, 3, 5, 16, 21, 24, 24, 21, 16, 61, 45, 24, 0, -24, -45, -61, -272, -333, -378, -402, -402, -378, -333, -272, -1385, -1113, -780, -402, 0, 402, 780, 1113, 1385, 7936, 9321, 10434, 11214, 11616, 11616, 11214, 10434, 9321, 7936

Offset: 0

N. J. A. Sloane, Feb 03 2019

The first row is a signed version of the Euler numbers A000111.

Other rows are defined by A(n+1,k) = A(n,k) + A(n,k+1).

			Triangular array T(n,k) = A(n-k,k) (n >= 0, k = 0..n), read from the antidiagonals upwards of square array A:
     0;
     1,    1;
     1,    0,   -1;
    -2,   -3,   -3,   -2;
    -5,   -3,    0,    3,    5;
    16,   21,   24,   24,   21,   16;
    61,   45,   24,    0,  -24,  -45,  -61;
  -272, -333, -378, -402, -402, -378, -333, -272;
  ...
From _Petros Hadjicostas_, Mar 04 2021: (Start)
Square array A(n,k) (n, k >= 0) begins:
   0,  1,   -1,   -2,     5,    16,     -61,    -272,     1385, ...
   1,  0,   -3,    3,    21,   -45,    -333,    1113,     9321, ...
   1, -3,    0,   24,   -24,  -378,     780,   10434,   -33264, ...
  -2, -3,   24,    0,  -402,   402,   11214,  -22830,  -480162, ...
  -5, 21,   24, -402,     0, 11616,  -11616, -502992,  1017600, ...
  16, 45, -378, -402, 11616,     0, -514608,  514608, 31880016, ...
  ... (End)

Alois P. Heinz, Antidiagonals n = 0..140, flattened
A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (matrix a_{n,k} on p. 18).

Cf. A000111, A002832 (next-to-main diagonal), A163747, A323834.

A323833 := proc(n,k)
    option remember;
    local i ;
    if k =0 then
        -A163747(n) ;
    elif n =0 then
        (-1)^k*A163747(k) ;
    elif k =n then
        0 ;
    else
        add(binomial(n,i)*procname(0,k+i), i=0..n) ;
    end if;
end proc:
seq(seq(A323833(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Jun 11 2025

{b(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+2, t=0; v = vector(k, i, if( i>1, t+= v[k+1-i]))); v[2])}; \\ Michael Somos's PARI program for A000111.
c(n) = if(n==0, 0, (-1)^floor(n/2)*b(n))
A(n, k) = sum(i=0, n, binomial(n, i)*c(k+i)) \\ Petros Hadjicostas, Mar 04 2021

From Petros Hadjicostas, Mar 04 2021: (Start)
Formulas about the square array A(n,k) (n,k > 0):
A(n,0) = -A163747(n) = (-1)^(n+1)*A(0,n) = if(n==0, 0, (-1)^floor(n/2)*A000111(n)).
A(n,n) = 0 and A(n,k) + (-1)^(n+k)*A(k,n) = 0.
A(n, k) = Sum_{i=0..n} binomial(n, i)*A(0,k+i).
Joint e.g.f.: Sum_{n,k >= 0} A(n,k)*(x^n/n!)*(y^k/k!) = 2*exp(-y)*(1 - exp(-x - y)) / (1 + exp(-2*(x + y))) = 2*exp(x)*(exp(x+y) - 1) / (exp(2*(x+y)) + 1).
Formulas about the triangular array T(n,k) = A(n-k,k) (0 <= k <= n):
T(n+1,k+1) = T(n+1,k) - T(n,k).
T(n,k) = -(-1)^n*T(n,n-k).
T(n,k) = Sum_{i=0..n-k} binomial(n-k,i)*T(k+i,k+i) for k=0..n with initial condition T(n,n) = (-1)^n*A163747(n). (End)

More terms from Alois P. Heinz, Feb 09 2019

cp's OEIS Frontend

A000940 Number of n-gons with n vertices.

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A000657 Median Euler numbers (the middle numbers of Arnold's shuttle triangle).

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A163747 Expansion of e.g.f. 2*exp(x)*(1-exp(x))/(1+exp(2*x)).

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A098277 Coefficients of polynomials D(n,x) related to median Euler numbers.

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A323833 A Seidel matrix A(n,k) read by antidiagonals upwards.

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A004577 Erroneous version of A000940.

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A163747 Expansion of e.g.f. 2exp(x)(1-exp(x))/(1+exp(2*x)).