A111594 -id:A111594 - cp's OEIS frontend

A000246 Number of permutations in the symmetric group S_n that have odd order.

1, 1, 1, 3, 9, 45, 225, 1575, 11025, 99225, 893025, 9823275, 108056025, 1404728325, 18261468225, 273922023375, 4108830350625, 69850115960625, 1187451971330625, 22561587455281875, 428670161650355625, 9002073394657468125, 189043541287806830625

Offset: 0

N. J. A. Sloane

Michael Reid (mreid(AT)math.umass.edu) points out that the e.g.f. for the number of permutations of odd order can be obtained from the cycle index for S_n, F(Y; X1, X2, X3, ... ) := e^(X1 Y + X2 Y^2/2 + X3 Y^3/3 + ... ) and is F(Y, 1, 0, 1, 0, 1, 0, ... ) = sqrt((1 + Y)/(1 - Y)).
a(n) appears to be the number of permutations on [n] whose up-down signature has nonnegative partial sums. For example, the up-down signature of (2,4,5,1,3) is (+1,+1,-1,+1) with nonnegative partial sums 1,2,1,2 and a(3)=3 counts (1,2,3), (1,3,2), (2,3,1). - David Callan, Jul 14 2006
This conjecture has been confirmed, see Bernardi, Duplantier, Nadeau link.
a(n) is the number of permutations of [n] for which all left-to-right minima occur in odd locations in the permutation. For example, a(3)=3 counts 123, 132, 231. Proof: For such a permutation of length 2n, you can append 1,2,..., or 2n+1 (2n+1 choices) and increase by 1 the original entries that weakly exceed the appended entry. This gives all such permutations of length 2n+1. But if the original length is 2n-1, you cannot append 1 (for then 1 would be a left-to-right min in an even location) so you can only append 2,3,..., or 2n (2n-1 choices). This count matches the given recurrence relation a(2n)=(2n-1)a(2n-1), a(2n+1)=(2n+1)a(2n). - David Callan, Jul 22 2008
a(n) is the n-th derivative of exp(arctanh(x)) at x = 0. - Michel Lagneau, May 11 2010
a(n) is the absolute value of the Moebius number of the odd partition poset on a set of n+1 points, where the odd partition poset is defined to be the subposet of the partition poset consisting of only partitions using odd part size (as well as the maximum element for n even). - Kenneth M Monks, May 06 2012
Number of permutations in S_n in which all cycles have odd length. - Michael Somos, Mar 17 2019
a(n) is the number of unranked labeled binary trees compatible with the binary labeled perfect phylogeny that, among possible two-leaf binary labeled perfect phylogenies for a sample of size n+2, is compatible with the smallest number of unranked labeled binary trees. - Noah A Rosenberg, Jan 16 2025

			For the Wallis numerators, denominators and partial products see A001900. - _Wolfdieter Lang_, Dec 06 2017

H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 87.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
Ron M. Adin, Pál Hegedűs, and Yuval Roichman, Descent set distribution for permutations with cycles of only odd or only even lengths, arXiv:2502.03507 [math.CO], 2025. See p. 2.
Joel Barnes, Conformal welding of uniform random trees, Ph. D. Dissertation, Univ. Washington, 2014.
Olivier Bernardi, Bertrand Duplantier and Philippe Nadeau, A Bijection Between Well-Labelled Positive Paths and Matchings, Séminaire Lotharingien de Combinatoire (2010), volume 63, Article B63e.
William Y. C. Chen, Breaking Cycles, the Odd Versus the Even, 2023.
A. Edelman and M. La Croix, The Singular Values of the GUE (Less is More), arXiv preprint arXiv:1410.7065 [math.PR], 2014-2015. See Table 1.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
A. Ghitza and A. McAndrew, Experimental evidence for Maeda's conjecture on modular forms, arXiv preprint arXiv:1207.3480 [math.NT], 2012.
Y. Cha, Closed form solutions of difference equations (2011) PhD Thesis, Florida State University, page 24
Dmitry Kruchinin, Integer properties of a composition of exponential generating functions, arXiv:1211.2100 [math.NT], 2012.
Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.
Kenneth M. Monks, An Elementary Proof of the Explicit Formula for the Möbius Number of the Odd Partition Poset, J. Int. Seq., Vol. 21 (2018), Article 18.9.6.
Julia A. Palacios, Anand Bhaskar, Filippo Disanto, and Noah A. Rosenberg, Enumeration of binary trees compatible with a perfect phylogeny, J. Math. Biol. 84 (2022), 54.
Qingchun Ren, Ordered Partitions and Drawings of Rooted Plane Trees, arXiv preprint arXiv:1301.6327 [math.CO], 2013. See Lemma 15.
Marko Riedel, et al. From combinatorial class to recurrence to closed form, Mathematics Stack Exchange.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
Jonathan Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
Sam Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv:1810.00993 [math.CO], 2018.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 108.
Allen Wang, Permutations with Up-Down Signatures of Nonnegative Partial Sums, MIT PRIMES Conference (2018).
David G. L. Wang and T. Zhao, The peak and descent statistics over ballot permutations, arXiv:2009.05973 [math.CO], 2020.
Index entries for sequences related to groups

Cf. A001900, A059838, A002867.
Bisections are A001818 and A079484.
Row sums of unsigned triangle A049218 and of A111594, A262125.
Main diagonal of A262124.
Cf. A002019.

a000246 n = a000246_list !! n
a000246_list = 1 : 1 : zipWith (+)
   (tail a000246_list) (zipWith (*) a000246_list a002378_list)
-- Reinhard Zumkeller, Feb 27 2012

I:=[1,1]; [n le 2 select I[n] else Self(n-1)+(n^2-5*n+6)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, May 02 2015

a:= proc(n) option remember; `if`(n<2, 1,
      a(n-1) +(n-1)*(n-2)*a(n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 14 2018

a[n_] := a[n] = a[n-1]*(n+Mod[n, 2]-1); a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 21 2011, after Pari *)
a[n_] := a[n] = (n-2)*(n-3)*a[n-2] + a[n-1]; a[0] := 0; a[1] := 1; Table[a[i], {i, 0, 20}] (* or *)  RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(n-2)*(n-3)a[n-2]+a[n-1]}, a, {n, 20}] (* G. C. Greubel, May 01 2015 *)
CoefficientList[Series[Sqrt[(1+x)/(1-x)], {x, 0, 20}], x]*Table[k!, {k, 0, 20}] (* Stefano Spezia, Oct 07 2018 *)

PARI
```
a(n)=if(n<1,!n,a(n-1)*(n+n%2-1))
```

Vec( serlaplace( sqrt( (1+x)/(1-x) + O(x^55) ) ) )

a(n)=prod(k=3,n,k+k%2-1) \\ Charles R Greathouse IV, May 01 2015

a(n)=(n!/(n\2)!>>(n\2))^2/if(n%2,n,1) \\ Charles R Greathouse IV, May 01 2015

E.g.f.: sqrt(1-x^2)/(1-x) = sqrt((1+x)/(1-x)).
a(2*k) = (2*k-1)*a(2*k-1), a(2*k+1) = (2*k+1)*a(2*k), for k >= 0, with a(0) = 1.
Let b(1)=0, b(2)=1, b(k+2)=b(k+1)/k + b(k); then a(n+1) = n!*b(n+2). - Benoit Cloitre, Sep 03 2002
a(n) = Sum_{k=0..floor((n-1)/2)} (2k)! * C(n-1, 2k) * a(n-2k-1) for n > 0. - Noam Katz (noamkj(AT)hotmail.com), Feb 27 2001
Also successive denominators of Wallis's approximation to Pi/2 (unreduced): 1/1 *  2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * .., for n >= 1.
D-finite with recurrence: a(n) = a(n-1) + (n-1)*(n-2)*a(n-2). - Benoit Cloitre, Aug 30 2003
a(n) is asymptotic to (n-1)!*sqrt(2*n/Pi). - Benoit Cloitre, Jan 19 2004
a(n) = n! * binomial(n-1, floor((n-1)/2)) / 2^(n-1), n > 0. - Ralf Stephan, Mar 22 2004
E.g.f.: e^atanh(x), a(n) = n!*Sum_{m=1..n} Sum_{k=m..n} 2^(k-m)*Stirling1(k,m) *binomial(n-1,k-1)/k!, n > 0, a(0)=1. - Vladimir Kruchinin, Dec 12 2011
G.f.: G(0) where G(k) = 1 + x*(4*k-1)/((2*k+1)*(x-1) - x*(x-1)*(2*k+1)*(4*k+1)/(x*(4*k+1) + 2*(x-1)*(k+1)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 24 2012
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+1)/(1-x/(x - 1/(1 - (2*k+1)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: G(0), where G(k) = 1 + x*(2*k+1)/(1 - x*(2*k+1)/(x*(2*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
For n >= 1, a(2*n) = (2*n-1)!!^2, a(2*n+1) = (2*n+1)*(2*n-1)!!^2. - Vladimir Shevelev, Dec 01 2013
E.g.f.: arcsin(x) - sqrt(1-x^2) + 1 for a(0) = 0, a(1) = a(2) = a(3) = 1. - G. C. Greubel, May 01 2015
Sum_{n>1} 1/a(n) = (L_0(1) + L_1(1))*Pi/2, where L is the modified Struve function. - Peter McNair, Mar 11 2022
From Peter Bala, Mar 29 2024: (Start)
a(n) =  n! * Sum_{k = 0..n} (-1)^(n+k)*binomial(1/2, k)*binomial(-1/2, n-k).
a(n) = (1/4^n) * (2*n)!/n! * hypergeom([-1/2, -n], [1/2 - n], -1).
a(n) = n!/2^n * A063886(n). (End)

A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 9, 0, 14, 0, 1, 0, 89, 0, 30, 0, 1, 225, 0, 439, 0, 55, 0, 1, 0, 3429, 0, 1519, 0, 91, 0, 1, 11025, 0, 24940, 0, 4214, 0, 140, 0, 1, 0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1, 893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1, 0

Offset: 0

Vladeta Jovovic, Apr 01 2001

The row polynomials t(n,x):=Sum_{k=0..n} T(n,k)*x^k satisfy the recurrence relation t(n,x) = x*t(n-1,x) + ((n-1)^2)*t(n-2,x); t(-1,x)=0, t(0,x)=1. - Wolfdieter Lang, see above.
This is an example of a Sheffer triangle (coefficient triangle for Sheffer polynomials). In the umbral calculus (see the Roman reference given under A048854) s(n,x) := Sum_{k=0..n} T(n,k)*x^k would be called Sheffer polynomials for (1/cosh(t),tanh(t)), which translates to the e.g.f. for column number k>=0 given by (1/sqrt(1-x^2))*((arctanh(x))^k)/k!. The e.g.f. given below is rewritten in this Sheffer context as (1/sqrt(1-x^2))*exp(y*log(sqrt((1+x)/(1-x))))= (1/sqrt(1-x^2))*exp(y*arctanh(x)). The rows of the Jabotinsky type triangle |A049218| provide the coefficients of the associated polynomials. - Wolfdieter Lang, Feb 24 2005
The solution of the differential-difference relation f(n+1,x)= (d/dx)f(n,x) + (n^2)*f(n-1,x), n >= 1, with inputs f(0,x) and f(1,x) = (d/dx)f(0,x) is f(n,x) = t(n,d_x)*f(0,x), with the differential operator d_x:=d/dx and the row polynomials t(n,x) defined above. This problem appears in a computation of thermo field dynamics where f(0,x)=1/cosh(x). See the triangle A060081. - Wolfdieter Lang, Feb 24 2005
The inverse of the Sheffer matrix T with elements T(n,k) is the Sheffer matrix A060081. - Wolfdieter Lang, Jul 22 2005
T(n,k)=0 if n-k= 1(mod 2), else T(n,k) = sum of M2(n,p), p from {1,...,A000041(n)} restricted to partitions with exactly k odd parts and any nonnegative number of even parts. For the M2-multinomial numbers in A-St order see A036039(n,p). - Wolfdieter Lang, Aug 07 2007

			Triangle begins:
  [1],
  [0, 1],
  [1, 0, 1],
  [0, 5, 0, 1],
  [9, 0, 14, 0, 1],
  [0, 89, 0, 30, 0, 1],
  [225, 0, 439, 0, 55, 0, 1],
  [0, 3429, 0, 1519, 0, 91, 0, 1],
  [11025, 0, 24940, 0, 4214, 0, 140, 0, 1],
  [0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1],
  [893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1],
  [0, 23941125, 0, 14466221, 0, 1467290, 0, 41778, 0, 385, 0, 1],
  ...
Signed version begins:
  [1],
  [0, 1],
  [-1, 0, 1],
  [0, -5, 0, 1],
  [9, 0, -14, 0, 1],
  [0, 89, 0, -30, 0, 1],
  [-225, 0, 439, 0, -55, 0, 1],
  [0, -3429, 0, 1519, 0, -91, 0, 1],
  ...
From _Peter Bala_, Feb 23 2024: (Start)
Maple can verify the following series for Pi:
Row 1 polynomial R(1, x) = x:
Pi = 3 + 4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(1, 2*n)*R(1, 2*n+2)).
Row 3 polynomial R(3, x) = 5*x + x^3:
(3/2)^2 * Pi = 7 + 4*(3^4)*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(3, 2*n)*R(3, 2*n+2)).
Row 5 polynomial R(5, x) = 89*x + 30*x^3 + x^5:
((3*5)/(2*4))^2 * Pi = 11 + 4*(3*5)^4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(5, 2*n)*R(5, 2*n+2)). (End)

Alois P. Heinz, Rows n = 0..140, flattened
Marcelo Aguiar, Sarah Brauner, and Vic Reiner, Configuration spaces and peak representations, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 13. See p. 12.
Peter Bala, Meixner polynomials and Brouncker's continued fraction for Pi
Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
Adel Hamdi and Jiang Zeng, Orthogonal polynomials and operator orderings, J. Math. Phys., 51:043506, (2010); arXiv:1006.0808 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Meixner polynomial of the first kind.

Cf. A060338, A060523, A094368, A028353 (col 1), A103916 (col 2), A103917 (col 3), A103918 (col 4).

Cf. A111594 (associated Sheffer polynomials), A142979, A142983.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1)*
      `if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 09 2015
# alternative
A060524 := proc(n,k)
    option remember;
    if nR. J. Mathar, Jul 06 2023

nn = 6; Range[0, nn]! CoefficientList[
   Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 28 2012 *)

E.g.f.: (1+x)^((y-1)/2)/(1-x)^((y+1)/2).
T(n, k) = T(n-1, k-1) + ((n-1)^2)*T(n-2, k); T(-1, k):=0, T(n, -1):=0, T(0, 0)=1, T(n, k)=0 if nWolfdieter Lang, see above.
The Meixner polynomials defined by S_0(x)=1, S_1(x) = x; S_{n+1}(x) = x*S_n(x) - n^2*S_{n-1}(x) give a signed version of this triangle (cf. A060338). - N. J. A. Sloane, May 30 2013
From Peter Bala, Apr 10 2024: (Start)
The n-th row polynomial R(n, x) satisfies
(4*n + 2)*R(n, x) = (x + 1)*R(n, x+2) - (x - 1)*R(n, x-2).
Series for Pi involving the row polynomials R(n, x): for n >= 0 there holds
((2*n + 1)!!/(2^n*n!))^2 * Pi = (4*n + 3) + 4*((2*n + 1)!!^4) * Sum_{k >= 1} (-1)^(k+1)/((2*k + 1)*R(2*n+1, 2*k)*R(2*n+1, 2*k+2)). Cf. A142979 and A142983.
R(2*n, 0) = A001147(n)^2 = A001818(n); R(2*n+1, 0) = 0.
R(n, 1) = n! = A000142(n).
R(2*n, 2) = (4*n + 1)*A001147(n)^2 = (4*n + 1)*((2*n)!/(2^n*n!))^2;
R(2*n+1, 2) = 2*A001447(n+1)^2 = 2*(2*n + 1)!^2/(n!^2*4^n).
R(n, 3) = (2*n + 1)*n! = A007680(n). (End)

A296676 Expansion of e.g.f. 1/(1 - arctanh(x)).

Original entry on oeis.org

1, 1, 2, 8, 40, 264, 2048, 18864, 196992, 2330112, 30519552, 440998656, 6940852224, 118501542912, 2177222879232, 42886017982464, 900748014944256, 20107190510714880, 475167358873239552, 11854636521914695680, 311291779253770911744, 8583598112533040332800, 247944624171011289907200

Offset: 0

Ilya Gutkovskiy, Dec 18 2017

nonn

			1/(1 - arctanh(x)) = 1 + x/1! + 2*x^2/2! + 8*x^3/3! + 40*x^4/4! + 264*x^5/5! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..430

Cf. A000246, A000828, A010050, A111594, A191700, A296675.

S:= series(1/(1-arctanh(x)),x,41):
seq(coeff(S,x,j)*j!,j=0..40); # Robert Israel, Dec 18 2017
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(j::odd,
      a(n-j)*binomial(n, j)*(j-1)!, 0), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 22 2021

nmax = 22; CoefficientList[Series[1/(1 - ArcTanh[x]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[1/(1 + (Log[1 - x] - Log[1 + x])/2), {x, 0, nmax}], x] Range[0, nmax]!

x='x+O('x^99); Vec(serlaplace(1/(1+(log(1-x)-log(1+x))/2))) \\ Altug Alkan, Dec 18 2017

E.g.f.: 1/(1 + (log(1 - x) - log(1 + x))/2).
a(n) ~ n! * 4*exp(2) * (exp(2)+1)^(n-1) / (exp(2)-1)^(n+1). - Vaclav Kotesovec, Dec 18 2017
a(n) = Sum_{k=0..n} k! * A111594(n,k). - Seiichi Manyama, Jun 30 2025

A111593 Triangle of tanh numbers.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, -2, 0, 1, 0, 0, -8, 0, 1, 0, 16, 0, -20, 0, 1, 0, 0, 136, 0, -40, 0, 1, 0, -272, 0, 616, 0, -70, 0, 1, 0, 0, -3968, 0, 2016, 0, -112, 0, 1, 0, 7936, 0, -28160, 0, 5376, 0, -168, 0, 1, 0, 0, 176896, 0, -135680, 0, 12432, 0, -240, 0, 1, 0, -353792, 0, 1805056, 0, -508640, 0, 25872

Offset: 0

Wolfdieter Lang, Aug 23 2005

Sheffer triangle associated to Sheffer triangle A060081.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
In the umbral calculus (see the S. Roman reference) this triangle would be called associated for (1,arctanh(y)).
Without the n=0 row and m=0 column and unsigned, this is the Jabotinsky triangle A059419.
The inverse matrix of A with elements a(n,m), n,m>=0, is A111594.
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060081, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*tanh(y)).
Exponential Riordan array [1, tanh(x)], inverse of [1, arctanh(x)] which is A111594. - Paul Barry, May 30 2010
Also the Bell transform of A155585(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			Binomial convolution of row polynomials: p(3,x)= -2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1, together with those from A060081:
s(3,x)= -5*x+x^3; s(2,x)= -1+x^2, s(1,x)= x, s(0,x)= 1;
therefore -5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = -2*y+y^3 + 3*x*y^2 + 3*(-1+x^2)*y + (-5*x+x^3).
From _Paul Barry_, May 30 2010: (Start)
Triangle begins:
  1;
  0,     1;
  0,     0,     1;
  0,    -2,     0,     1;
  0,     0,    -8,     0,     1;
  0,    16,     0,   -20,     0,     1;
  0,     0,   136,     0,   -40,     0,     1;
  0,  -272,     0,   616,     0,   -70,     0,     1;
  0,     0, -3968,     0,  2016,     0,  -112,     0,     1;
Production matrix begins:
  0,   1;
  0,   0,   1;
  0,  -2,   0,   1;
  0,   0,  -6,   0,   1;
  0,   0,   0, -12,   0,   1;
  0,   0,   0,   0, -20,   0,   1;
  0,   0,   0,   0,   0, -30,   0,   1;
  0,   0,   0,   0,   0,   0, -42,   0,   1;
  0,   0,   0,   0,   0,   0,   0, -56,   0,   1; (End)

W. Lang, First 10 rows.

Row sums: A003723. Unsigned row sums: A006229.
Cf. A059419, A060081, A111594.
Cf. A002378.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> 2^(n+1)*euler(n+1, 1), 9); # Peter Luschny, Jan 26 2016

t[0, 0] = 1; t[n_, m_] := Sum[ Binomial[k+m-1, m-1]*(k+m)!*(-1)^(k)*2^(n-k-m)*StirlingS2[n, k+m], {k, 0, n-m}]/m!; Table[t[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[2^(#+1)*EulerE[#+1, 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

T(n,m):=if n=0 and m=0 then 1 else sum(binomial(k+m-1,m-1)*(k+m)!*(-1)^(k)*2^(n-k-m)*stirling2(n,k+m),k,0,n-m)/m!; /* Vladimir Kruchinin, Jun 09 2011 */

# uses[riordan_array from A256893]
riordan_array(1, tanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015

E.g.f. for column m>=0: ((tanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((tanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) - (m+1)*m*a(n-1, m+1), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for nT(n,m) = (Sum_{k=0..n-m} binomial(k+m-1,m-1)*(k+m)!*(-1)^k*2^(n-k-m)*stirling2(n,k+m))/m!, T(0,0)=1. - Vladimir Kruchinin, Jun 09 2011
With e.g.f. exp(x*tanh(t)) = sum(n>= 0, P(n,x)*t^n/n!), the lowering operator is L = arctanh(d/dx) = d/dx + (1/3)(d/dx)^3 + (1/5)(d/dx)^5 + ..., and the raising operator is R = x [1 - (d/dx)^2], where L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x), since the sequence is a binomial Sheffer sequence. - Tom Copeland, Oct 01 2015
The raising operator R = x - x D^2 in matrix form acting on an o.g.f. (formal power series) is the transpose of the production matrix M below. The linear term x is the diagonal of ones after transposition. The other transposed diagonal (A002378) comes from -x D^2 x^n = -n * (n-1) x^(n-1). Then P(n,x) = (1,x,x^2,..) M^n (1,0,0,..)^T. - Tom Copeland, Aug 17 2016

A385468 Expansion of e.g.f. 1/(1 - 2 * arctanh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 17, 129, 1269, 15147, 213765, 3475329, 64020585, 1317472563, 29960707545, 746086414785, 20192521440285, 590166330458715, 18525204423695565, 621571306435103745, 22199954036873457105, 840913892465144800995, 33672216851574639900705

Offset: 0

Seiichi Manyama, Jun 30 2025

Cf. A296676, A385469.

Cf. A001147, A111594, A385376.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*atanh(x))^(1/2)))

E.g.f.: 1/(1 - log((1+x)/(1-x)))^(1/2).
a(n) = Sum_{k=0..n} A001147(k) * A111594(n,k).
a(n) ~ 2 * (exp(1) + 1)^(n - 1/2) * n^n / (exp(n - 1/2) * (exp(1) - 1)^(n + 1/2)). - Vaclav Kotesovec, Jun 30 2025

A385469 Expansion of e.g.f. 1/(1 - 3 * arctanh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 30, 312, 4224, 70176, 1384032, 31590912, 819254016, 23792039424, 764912590848, 26970073390080, 1034798724320256, 42921327875788800, 1913760046417508352, 91281373260924026880, 4637755280044146032640, 250054580636566927441920, 14259891701316651909120000

Offset: 0

Seiichi Manyama, Jun 30 2025

nonn

Cf. A296676, A385468.

Cf. A007559, A111594.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*atanh(x))^(1/3)))

E.g.f.: 1/(1 - (3/2) * log((1+x)/(1-x)))^(1/3).

a(n) = Sum_{k=0..n} A007559(k) * A111594(n,k).

A385500 Expansion of e.g.f. exp( -LambertW(-arctanh(x)) ).

Original entry on oeis.org

1, 1, 3, 18, 149, 1640, 22359, 366128, 6998697, 153191808, 3779353515, 103800229632, 3141633970749, 103904351855616, 3728602377979647, 144297781732300800, 5991021498320041809, 265636734347975688192, 12527923794824003280723, 626224876080360687599616

Offset: 0

Seiichi Manyama, Jul 01 2025

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A385501, A385502.

Cf. A111594, A385425.

nmax=19; CoefficientList[Series[Exp[ -LambertW[-ArcTanh[x]]],{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Jul 01 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-atanh(x)))))

E.g.f. A(x) satisfies A(x) = exp( arctanh(x) * A(x) ).
E.g.f. A(x) satisfies A(x) = ( (1+x)/(1-x) )^(A(x)/2).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A111594(n,k).
a(n) ~ sqrt(exp(4*exp(-1)) - 1) * n^(n-1) / (2*exp(n - 3/2 + exp(-1)) * tanh(exp(-1))^n). - Vaclav Kotesovec, Jul 01 2025

A385501 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-arctanh(x)) ).

Original entry on oeis.org

1, 1, 3, 18, 165, 2040, 31815, 599760, 13268745, 337115520, 9674678475, 309554784000, 10927053262125, 421849524096000, 17682153623909775, 799730490214656000, 38820939579369572625, 2013202580708487168000, 111081054630965602057875, 6497703571257963896832000

Offset: 0

Seiichi Manyama, Jul 01 2025

nonn

Cf. A385500, A385502.

Cf. A001147, A111594, A138020, A227466, A385426.

nmax=20; CoefficientList[(1/x) *InverseSeries[Series[x * Exp[-ArcTanh[x]],{x,0,nmax}],x] ,x]Range[0,nmax-1]! (* Stefano Spezia, Jul 01 2025 *)

a(n) = n!*sum(k=0, n, binomial(n, k)*binomial(n/2+k+1/2, n)/(n+2*k+1));

E.g.f. A(x) satisfies A(x) = exp( arctanh(x*A(x)) ).
E.g.f. A(x) satisfies A(x) = sqrt( (1+x*A(x))/(1-x*A(x)) ).
a(n) = Sum_{k=0..n} (n+1)^(k-1) * A111594(n,k).
a(n) = n!/2^n * A138020(n) = n! * Sum_{k=0..n} binomial(n,k) * binomial(n/2+k+1/2,n)/(n+2*k+1).

A385502 E.g.f. A(x) satisfies A(x) = exp( arctanh(x * A(x)) / A(x) ).

Original entry on oeis.org

1, 1, 1, 3, 25, 205, 2001, 25991, 394353, 6718041, 130319745, 2833146987, 67767170505, 1772434086501, 50392083769041, 1546052750636655, 50905035315373281, 1790951445870568113, 67050161599822764417, 2661363261252799648083, 111637709182606749500025

Offset: 0

Seiichi Manyama, Jul 01 2025

nonn

Cf. A385500, A385501.

Cf. A111594, A385428.

terms = 21; A[] = 1; Do[A[x] = Exp[ArcTanh[x*A[x]]/A[x]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 01 2025 *)

a111594(n, k) = my(x='x+O('x^(n+1))); n!*polcoef(atanh(x)^k/k!, n);
a(n) = sum(k=0, n, (n-k+1)^(k-1)*a111594(n, k));

E.g.f. A(x) satisfies A(x) = ( (1+x*A(x))/(1-x*A(x)) )^(1/(2*A(x))).
a(n) = Sum_{k=0..n} (n-k+1)^(k-1) * A111594(n,k).
a(n) ~ s * sqrt((-1 + r^2*s^2)/(1 - 2*r^2*(1 + r)*s^2 + r^4*s^4)) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.4422573061236400123439455590007131605377062990202... and s = 1.93686591146053883124948614770176661449449740697... are the roots of the system of equations ((1 + r*s)/(1 - r*s))^(1/2/s) = s, 2*r*s + (-1 + r^2*s^2)*(log((1 + r*s)/(1 - r*s)) + 2*s) = 0. - Vaclav Kotesovec, Jul 01 2025

A385470 Expansion of e.g.f. 1/(1 - 2 * arctanh(x)).

Original entry on oeis.org

1, 2, 8, 52, 448, 4848, 62912, 952992, 16496640, 321282816, 6952332288, 165489858048, 4297340166144, 120890184308736, 3662409013420032, 118879239686541312, 4115985952586858496, 151415632063102648320, 5897814669785134006272, 242489327746828076974080

Offset: 0

Seiichi Manyama, Jun 30 2025

Cf. A296676, A385471.

Cf. A111594, A385468.

With[{nn=20},CoefficientList[Series[1/(1-2ArcTanh[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 04 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*atanh(x))))

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A385468.
E.g.f.: 1/(1 - log((1+x)/(1-x))).
a(n) = Sum_{k=0..n} 2^k * k! * A111594(n,k).
a(n) ~ 2^(3/2) * sqrt(Pi) * (1 + exp(1))^(n-1) * n^(n + 1/2) / (exp(n-1) * (exp(1) - 1)^(n+1)). - Vaclav Kotesovec, Jun 30 2025

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cp's OEIS Frontend

A000246 Number of permutations in the symmetric group S_n that have odd order.

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A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0.

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A296676 Expansion of e.g.f. 1/(1 - arctanh(x)).

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A111593 Triangle of tanh numbers.

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A385468 Expansion of e.g.f. 1/(1 - 2 * arctanh(x))^(1/2).

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A385469 Expansion of e.g.f. 1/(1 - 3 * arctanh(x))^(1/3).

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A385500 Expansion of e.g.f. exp( -LambertW(-arctanh(x)) ).

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