A049218 -id:A049218 - 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)

A111594 Triangle of arctanh numbers.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 24, 0, 20, 0, 1, 0, 0, 184, 0, 40, 0, 1, 0, 720, 0, 784, 0, 70, 0, 1, 0, 0, 8448, 0, 2464, 0, 112, 0, 1, 0, 40320, 0, 52352, 0, 6384, 0, 168, 0, 1, 0, 0, 648576, 0, 229760, 0, 14448, 0, 240, 0, 1

Offset: 0

Wolfdieter Lang, Aug 23 2005

Sheffer triangle associated to Sheffer triangle A060524.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
The inverse matrix of A with elements a(n,m), n,m>=0, is given in A111593.
In the umbral calculus notation (see the S. Roman reference) this triangle would be called associated to (1,tanh(y)).
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060524 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.
Without the n=0 row and m=0 column and signed, this will become the Jabotinsky triangle A049218 (arctan numbers). For Jabotinsky matrices see the Knuth reference under A039692.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*arctanh(y)).
Exponential Riordan array [1, arctanh(x)] = [1, log(sqrt((1+x)/(1-x)))]. - Paul Barry, Apr 17 2008
Also the Bell transform of A005359. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 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 A060524:
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).
Triangle begins:
  1;
  0,   1;
  0,   0,    1;
  0,   2,    0,   1;
  0,   0,    8,   0,    1;
  0,  24,    0,  20,    0,  1;
  0,   0,  184,   0,   40,  0,   1;
  0, 720,    0, 784,    0, 70,   0, 1;
  0,   0, 8448,   0, 2464,  0, 112, 0, 1;
...

Wolfdieter Lang, First 10 rows.

Row sums: A000246.

Cf. A005359, A049218, A060524, A111593.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::even, n!, 0), 10); # Peter Luschny, Jan 27 2016

rows = 10;
t = Table[If[EvenQ[n], n!, 0], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

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

E.g.f. for column m>=0: ((arctanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((arctanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) + (n-2)*(n-1)*a(n-2, m), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n

A002019 a(n) = a(n-1) - (n-1)(n-2)a(n-2).

Original entry on oeis.org

1, 1, 1, -1, -7, 5, 145, -5, -6095, -5815, 433025, 956375, -46676375, -172917875, 7108596625, 38579649875, -1454225641375, -10713341611375, 384836032842625, 3663118565923375, -127950804666254375, -1519935859717136875

Offset: 0

N. J. A. Sloane

Dwight, Tables of Integrals ..., Eq. 552.5, page 133.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
G. Guillotte and L. Carlitz, Problem H-216 and solution, Fib. Quarter. p. 90, Vol 13, 1, Feb. 1975.
R. Kelisky, The numbers generated by exp(arctan x), Duke Math. J., 26 (1959), 569-581.
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Bisections are A102058 and A102059.
Cf. A006228.
Row sums of signed triangle A049218.
Cf. A000246.

a002019 n = a002019_list !! n
a002019_list = 1 : 1 : zipWith (-)
   (tail a002019_list) (zipWith (*) a002019_list a002378_list)
-- Reinhard Zumkeller, Feb 27 2012

I:=[1,1]; [1] cat [ n le 2 select I[n] else Self(n-1)-(n^2-3*n+2)*Self(n-2): n in [1..35]]; // Vincenzo Librandi, May 02 2015

RecurrenceTable[{a[0]==1,a[1]==1,a[n]==a[n-1]-(n-1)(n-2)a[n-2]}, a[n],{n,30}] (* Harvey P. Dale, May 02 2011 *)
CoefficientList[Series[E^(ArcTan[x]),{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Nov 06 2014 *)

a(n):=n!*sum(if oddp(m+n) then 0 else (-1)^((3*n+m)/2)/(2^m*m!)*sum(2^i*binomial(n-1,i-1)*m!/i!*stirling1(i,m),i,m,n),m,1,n); /* Vladimir Kruchinin, Aug 05 2010 */

E.g.f.: exp(arctan(x)).
a(n) = n!*sum(if oddp(m+n) then 0 else (-1)^((3*n+m)/2)/(2^m*m!)*sum(2^i*binomial(n-1,i-1)*m!/i!*stirling1(i,m),i,m,n),m,1,n), n>0. - Vladimir Kruchinin, Aug 05 2010
E.g.f.: exp(arctan(x)) = 1 + 2x/(H(0)-x); H(k) = 4k + 2 + x^2*(4k^2 + 8k + 5)/H(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2011
a(n+1) = a(n) - a(n-1) * A002378(n-2). - Reinhard Zumkeller, Feb 27 2012
E.g.f.: -2i*(B((1+ix)/2; (2-i)/2, (2+i)/2) - B(1/2; (2-i)/2, (2+i)/2)), for a(0)=0, a(1)=a(2)=a(3)=1, B(x;a,b) is the incomplete Beta function. - G. C. Greubel, May 01 2015
a(n) = i^n*n!*Sum_{r+s=n} (-1)^s*binomial(-i/2, r)*binomial(i/2,s) where i is the imaginary unit. See the Fib. Quart. link. - Michel Marcus, Jan 22 2017

More terms from Herman P. Robinson

A137513 Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.

Original entry on oeis.org

1, 0, 2, 0, 0, 4, 0, 4, 0, 8, 0, 0, 32, 0, 16, 0, 48, 0, 160, 0, 32, 0, 0, 736, 0, 640, 0, 64, 0, 1440, 0, 6272, 0, 2240, 0, 128, 0, 0, 33792, 0, 39424, 0, 7168, 0, 256, 0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512, 0, 0, 2594304, 0, 3676160, 0, 924672, 0, 61440, 0, 1024

Offset: 1

Roger L. Bagula, Apr 23 2008

Previous name was: Triangle read by rows: coefficients of the expansion of a polynomial related to the Poisson kernel: p(t,r) = ((1 + t)/(1 - t))^x: r*Exp(i*theta) -> t.
General relation is that Poisson's kernel is the real part of this type of function (page 31 Hoffman reference).
The row polynomials of this table are the Mittag-Leffler polynomials M(n,t), a polynomial sequence of binomial type [Roman, Chapter 4, Section 1.6]. The first few values are M(0,t) = 1, M(1,t) = 2*t, M(2,t) = 4*t^2, M(3,t) = 4*t+8*t^3. The polynomials M(n,t/2) are the (unsigned) row polynomials of A049218. - Peter Bala, Dec 04 2011
Also the Bell transform of the sequence "a(n) = 2*n! if n is even else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			{1},
{0, 2},
{0, 0, 4},
{0, 4, 0, 8},
{0, 0, 32, 0, 16},
{0, 48, 0, 160, 0, 32},
{0, 0, 736, 0, 640, 0, 64},
{0, 1440, 0, 6272, 0, 2240, 0, 128},
{0, 0, 33792, 0, 39424, 0, 7168, 0, 256},
{0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512},
{0, 0, 2594304, 0, 3676160,0, 924672, 0, 61440, 0, 1024}

Kenneth Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1962, page 30.
Thomas McCullough, Keith Phillips, Foundations of Analysis in the Complex Plane, Holt, Reinhart and Winston, New York, 1973, 215.
S. Roman, The Umbral Calculus: Dover Publications, New York (2005).

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
H. Bateman, The Polynomial of Mittag-Leffler. PNAS, 26 (8), 1940, 491-496.
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial

Cf. A049218, A098558 (row sums).

A137513_row := proc(n) `if`(n=0,1,2*x*hypergeom([1-n,1-x],[2],2));
PolynomialTools[CoefficientList](expand(n!*simplify(%,hypergeom)),x) end:
seq(A137513_row(n),n=0..10): ListTools[FlattenOnce]([%]); # Peter Luschny, Jan 28 2016
# Alternatively, using the function BellMatrix defined in A264428:
BellMatrix(n -> `if`(n::odd, 0, 2*n!), 9); # Peter Luschny, Jan 28 2016

p[t_] = ((1 + t)/(1 - t))^x; Table[ ExpandAll[n! * SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30} ]], n], x], {n, 0, 10}]; Flatten[a]
MLP[n_] := Sum[Binomial[n, k]*2^k*FactorialPower[n - 1, n - k]* FactorialPower[x, k] // FunctionExpand, {k, 0, n}]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)
MLP[0] = 1; MLP[n_] := 2x*n!*Hypergeometric2F1[1-n, 1-x, 2, 2]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)
BellMatrix[If[OddQ[#], 0, 2*#!]&, 9] (* in triangular matrix form, using Peter Luschny's BellMatrix function defined in A264428 *) (* Jean-François Alcover, Jan 29 2016 *)

MLP = lambda n: sum(binomial(n, k)*2^k*falling_factorial(n-1, n-k)* falling_factorial(x, k) for k in (0..n)).expand()
def A137513_row(n): return MLP(n).list()
for n in (0..9): A137513_row(n) # Peter Luschny, Jan 28 2016

From Peter Bala, Dec 04 2011: (Start)
T(n,k) = (-1)^k*(n-1)!*Sum_{i=k..n} (-2)^i*binomial(n,i)/(i-1)!*|Stirling1(i,k)|.
E.g.f.: Sum_{n>=0} M(n,t)*x^n/n! = exp(t*log((1+x)/(1-x))) = ((1+x)/(1-x))^t = exp(2*t*atanh(x)) = 1 + (2*t)*x + (4*t^2)*x^2/2! + (4*t+8*t^3)*x^3/3! + ....
M(n,t) = (n-1)!*Sum_{k = 1..n} k*2^k*binomial(n,k)*binomial(t,k), for n>=1.
Recurrence relation: M(n+1,t) = 2*t*Sum_{k = 0..floor(n/2)} (n!/(n-2*k)!)* M(n-2*k,t), with M(0,t) = 1.
The o.g.f. for the n-th diagonal of the table is a rational function in t, given by the coefficient of x^n/n! in the expansion (with respect to x) of the compositional inverse (x-t*log((1+x)/(1-x)))^(-1) = x/(1-2*t) + 4*t/(1-2*t)^4*x^3/3! + (48*t+64*t^2)/(1-2*t)^7*x^5/5! + ...; for example, the o.g.f. for the fifth subdiagonal is (48*t+64*t^2)/(1-2*t)^7 = 48*t + 736*t^2 + 6272*t^3+ .... See the Bala link.
(End)
The row polynomials satisfy M(n, t+1) - M(n, t-1) = 2*n*M(n, t)/t. - Peter Bala, Nov 16 2016

Edited and new name by Peter Luschny, Jan 28 2016

A002429 Numerators of double sums of reciprocals.

Original entry on oeis.org

1, 1, 14, 818, 141, 13063, 16774564, 1057052, 4651811, 778001383, 1947352646, 1073136102266, 72379420806883, 112229882767, 120372921248744, 13224581478608216, 2077531074698521033, 517938126297258811, 13785854249175914469406, 343586489824688536178, 1958290344469311726833

Offset: 0

N. J. A. Sloane

nonn

Also, numerators of coefficients of expansion of arctan(x)^3. - Ruperto Corso, Dec 09 2011

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 117.
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).

G. C. Greubel, Table of n, a(n) for n = 0..770
Mohammad K. Azarian, A Double Sum, Problem 440, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. Solution published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.
H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 316.

Cf. A008309, A049218.

List([0..25], n-> NumeratorRat( 3*Sum([3..2*n+3], k-> (-1)^(k-1)*2^(k-2)* Binomial(2*(n+1),k-1)*Stirling1(k,3)/Factorial(k)) )) # G. C. Greubel, Jul 03 2019

[Numerator(3*(&+[2^(k-2)*Binomial(2*(n+1), k-1)* StirlingFirst(k,3)/Factorial(k): k in [3..2*n+3]]) ): n in [0..25]]; // G. C. Greubel, Jul 03 2019

p2x:=proc(n) option remember: if(n=1) then RETURN(1) else RETURN(((n-1)*p2x(n-1)+1/(2*n-1))/n) fi: end proc;
p3x:=proc(n) option remember: if(n=1) then RETURN(1) else RETURN(((2*n-1)*p3x(n-1)+3*p2x(n))/(2*n+1)) fi: end proc;
A002429 := proc(n)
    numer(p3x(n)) ;
end proc:
seq(A002429(n),n=1..25) ; # Ruperto Corso, Dec 09 2011

a[n_]:= (-1)^n*SeriesCoefficient[ArcTan[x]^3, {x, 0, 2*n+3}]//Numerator; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 04 2013 *)
a[n_]:= Numerator[3*Sum[2^(k-2)*Binomial[2*(n+1),k-1]*StirlingS1[k,3]/k!, {k,3,2*n+3}]]; Table[a[n], {n,0,25}] (* G. C. Greubel, Jul 03 2019 *)

stirling1(n, k)=if(n<1, 0, n!*polcoeff(binomial(x, n), k))
for(n=0,25,print1(numerator(3/4*sum(i=3,2*n+3,2^i*binomial(2*(n+1),i-1)*stirling1(i,3)/ i!))",")) \\ Ruperto Corso, Dec 09 2011

[numerator( 3*sum((-1)^(k-1)*2^(k-2)*binomial(2*(n+1), k-1)* stirling_number1(k,3)/factorial(k) for k in (3..2*n+3)) ) for n in (0..25)] # G. C. Greubel, Jul 03 2019

a(n) = numerator of 3*Sum_{i=3..2*n+3} 2^(i-2)*binomial(2*(n+1),i-1) *Stirling1(i,3)/ i!. - Ruperto Corso, Dec 09 2011

More terms from Ruperto Corso, Dec 09 2011

A008309 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

Original entry on oeis.org

1, 1, -2, 1, -8, 1, 24, -20, 1, 184, -40, 1, -720, 784, -70, 1, -8448, 2464, -112, 1, 40320, -52352, 6384, -168, 1, 648576, -229760, 14448, -240, 1, -3628800, 5360256, -804320, 29568, -330, 1

Offset: 1

N. J. A. Sloane

			With the zero coefficients included the data begins 1; 0,1; -2,0,1; 0,-8,0,1; 24,0,-20,0,1; 0,184,0,-40,0,1; ..., which is A049218.
The table without zeros begins
    1;
    1;
   -2,   1;
   -8,   1;
   24, -20,   1;
  184, -40,   1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.

Essentially same as A049218.
A007290(n) = -T(n, floor(n-1)/2);
A010050(n) = (-1)^n*T(2n+1, 1);
A049034(n) = (-1)^n*T(2n+2, 1);
A049214(n) = (-1)^n*T(2n+3, 2);
A049215(n) = (-1)^n*T(2n+4, 2);
A049216(n) = (-1)^n*T(2n+5, 3);
A049217(n) = (-1)^n*T(2n+6, 3).

t[n_, k_] := (-1)^((3*n+k)/2)*n!/2^k*Sum[2^i*Binomial[n-1, i-1]*StirlingS1[i, k]/i!, {i, k, n}]; Flatten[Table[t[n,k], {n,1,11}, {k, 2-Mod[n, 2], n, 2}]] (* Jean-François Alcover, Aug 31 2011, after Vladimir Kruchinin *)

a(n)=atan(x)^n/n!
T(n,k)=polcoeff(serlaplace(a(2*k-n%2)), n)

E.g.f.: arctan(x)^k/k! = Sum_{n>=0} T(m, floor((k+1)/2))* x^m/m!, where m = 2*n + k mod 2.

Additional comments from Michael Somos

A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.

Original entry on oeis.org

1, 0, 1, 0, 6, 3, 0, 120, 90, 15, 0, 5040, 4620, 1260, 105, 0, 362880, 378000, 132300, 18900, 945, 0, 39916800, 45571680, 18711000, 3534300, 311850, 10395, 0, 6227020800, 7628100480, 3511347840, 794593800, 94594500, 5675670, 135135

Offset: 0

José H. Nieto S., Jan 15 2012

The row polynomials t(n,x):= sum(T(n,k)*x^k, k=0..n) satisfy the recurrence relation t(n,x) = (2n-1)*(x+2n-2)*t(n-1,x), with t(0,x)=1, hence t(n,x)=(2n-1)!!*x(x+2)(x+4)...(x+2n-2).

			1;
0,        1,
0,        6,        3;
0,      120,       90,       15;
0,     5040,     4620,     1260,     105;
0,   362880,   378000,   132300,   18900,    945;
0, 39916800, 45571680, 18711000, 3534300, 311850, 10395;

Alois P. Heinz, Rows n = 0..85, flattened
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Angelo Lucia and Amanda Young, A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs, arXiv:2212.11872 [math-ph], 2022.

Cf. A049218, A060523, A060524, A132393, A048994.

Row sums give: A001818. - Alois P. Heinz, Jul 21 2013

T_row:= proc(n) local k; seq(doublefactorial(2*n-1)*2^(n-k)* coeff(expand(pochhammer(x, n)), x, k), k=0..n) end: seq(T_row(n), n=0..10);

nn=12;Prepend[Map[Prepend[Select[#,#>0&],0]&,Table[(Range[0, nn]!CoefficientList[ Series[(1-x^2)^(-y/2),{x,0,nn}],{x,y}])[[n]],{n,3,nn,2}]],{1}]//Grid (* Geoffrey Critzer, Jul 21 2013 *)

T(n,k) = (2*n)!/(2^n*n!)*(-2)^(n-k)*stirling(n,k,1); \\ Andrew Howroyd, Feb 12 2018

T(n,k) = (2n-1)!!*2^(n-k)*A132393(n,k).
T(n,k) = (2n-1)T(n-1,k-1) + (2n-1)(2n-2)*T(n-1,k); T(0,0)=1, T(n,0)=0 for n>0,
T(n,n) = (2n-1)!! = A001147(n).
T(n,1) = (2n-1)! = A009445(n-1).

Showing 1-8 of 8 results.

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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A111594 Triangle of arctanh numbers.

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A002019 a(n) = a(n-1) - (n-1)(n-2)a(n-2).

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A137513 Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.

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A002429 Numerators of double sums of reciprocals.

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