A111595 -id:A111595 - cp's OEIS frontend

A001818 Squares of double factorials: (135*...(2n-1))^2 = ((2n-1)!!)^2.

1, 1, 9, 225, 11025, 893025, 108056025, 18261468225, 4108830350625, 1187451971330625, 428670161650355625, 189043541287806830625, 100004033341249813400625, 62502520838281133375390625, 45564337691106946230659765625, 38319607998220941779984862890625

Offset: 0

N. J. A. Sloane

Number of permutations in S_{2n} in which all cycles have even length (cf. A087137).
Also number of permutations in S_{2n} in which all cycles have odd length. - Vladeta Jovovic, Aug 10 2007
a(n) is the sum over all multinomials M2(2*n,k), k from {1..p(2*n)} restricted to partitions with only even parts. p(2*n)= A000041(2*n) (partition numbers) and for the M2-multinomial numbers in A-St order see A036039(2*n,k). - Wolfdieter Lang, Aug 07 2007
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture 1: For any primitive 2n-th root zeta of unity, the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} coincides with a(n) = ((2n-1)!!)^2, where m(j,k) is (1+zeta^(j-k))/(1-zeta^(j-k)) if j is not equal to k, and 1 otherwise.
The determinant of [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n-1)!!)^2/(2n-1) by Han Wang and Zhi-Wei Sun in 2022.
Conjecture 2: Let p be an odd prime. Then the permanent of (p-1) X (p-1) matrix [f(j,k)]_{j,k=1..p-1} is congruent to a((p-1)/2) = ((p-2)!!)^2 modulo p^2, where f(j,k) is (j+k)/(j-k) if j is not equal to k, and f(j,k) = 1 otherwise. (End)

			Multinomial representation for a(2): partitions of 2*2=4 with even parts only: (4) with position k=1, (2^2) with k=3; M2(4,1)= 6 and M2(4,3)= 3, adding up to a(2)=9.
G.f. = 1 + x + 9*x^2 + 225*x^3 + 11025*x^4 + 893025*x^5 + 108056025*x^6 + ...

John Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.34(c).

T. D. Noe, Table of n, a(n) for n = 0..50
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.
David Callan and Emeric Deutsch, The Run Transform, arXiv preprint arXiv:1112.3639 [math.CO], 2011.
William Y. C. Chen and Elena L. Wang, r-Enriched Permutations and an Inequality of Bóna-McLennan-White, arXiv:2502.04136 [math.CO], 2025. See p. 4.
Harry Crane and Peter McCullagh, Reversible Markov structures on divisible set partitions, Journal of Applied Probability, Vol. 52, No. 3 (2015), pp. 622-635.
Muhammad Adam Dombrowski and Gregory Dresden, Areas Between Cosines, arXiv:2404.17694 [math.CO], 2024. See p. 11.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 6.
IBM, "Ponder This" puzzle for June 2009. [From _Vladeta Jovovic_, Jul 26 2009]
John Riordan and N. J. A. Sloane, Correspondence, 1974.
Terence Tao, A differentiation identity.
Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Struve function.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
Index to divisibility sequences.

Cf. A001147, A002454, A111595, A197037.
Bisection of A012248.
Right-hand column 1 in triangle A008956.

DoubleFactorial:=func< n | &*[n..2 by -2] >; [DoubleFactorial((2*n-1))^2: n in [0..20] ]; // Vincenzo Librandi, Jul 21 2017

a := proc(m) local k; 4^m*mul((-1)^k*(k-m-1/2),k=1..2*m) end; # Peter Luschny, Jun 01 2009

FoldList[Times,1,Range[1,25,2]]^2 (* or *) Join[{1},(Range[1,29,2]!!)^2] (* Harvey P. Dale, Jun 06 2011, Apr 10 2012 *)
Table[((2 n - 1)!!)^2, {n, 0, 30}] (* Vincenzo Librandi, Jul 21 2017 *)

PARI
```
a(n)=((2*n)!/(n!*2^n))^2
```

{a(n) = if( n<0, 1 / a(-n), sqr((2*n)! / (n! * 2^n)))}; /* Michael Somos, Jan 06 2017 */

a(n) = A001147(n)^2.
a(n) = A111595(2*n, 0).
a(n) = (2*n-1)!*Sum_{k=0..n-1} binomial(2*k,k)/4^k, n >= 1. - Wolfdieter Lang, Aug 23 2005
arcsinh(x) = Sum_{n>=1} (-1)^(n-1)*a(n)*x^(2*n-1)/(2*n-1)!. - James R. Buddenhagen, Mar 24 2009
From Karol A. Penson, Oct 21 2009: (Start)
G.f.: Sum_{n>=0} a(n)*x^n/(n!)^2 = 2*EllipticK(2*sqrt(x))/Pi.
Asymptotically: a(n) = (2/((exp(-1/2))^2*(exp(1/2))^2)-1/(6*(exp(-1/2))^2*(exp(1/2))^2*n)+1/(144*(exp(-1/2))^2*(exp(1/2))^2*n^2)+O(1/n^3))*(2^n)^2/(((1/n)^n)^2*(exp(n))^2), n->infinity.
Integral representation as n-th moment of a positive function on a positive halfaxis (solution of the Stieltjes moment problem), in Maple notation:
a(n) = Integral_{x>=0} x^n*BesselK(0,sqrt(x))/(Pi*sqrt(x)).
This solution is unique.
(End)
D-finite with recurrence: a(0) = 1, a(n) = (2*n-1)^2*a(n-1), n > 0.
a(n) ~ 2*2^(2*n)*e^(-2*n)*n^(2*n). - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: 1/sqrt(1-x^2) = Sum_{n >= 0} a(n)*x^(2*n)/(2*n)!. Also arcsin(x) = Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)!. - Michael Somos, Jul 03 2002
(-1)^n*a(n) is the coefficient of x^0 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22 2002
-arccos(x) + Pi/2 = x + x^3/3! + 9*x^5/5! + 225*x^7/7! + 11205*x^9/9! + ... - Tom Copeland, Oct 23 2008
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) =  1 - (4*k^2+4*k+1)/(1-x/(x - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
a(n) = det(V(i+1,j), 1 <= i,j <= n), where V(n,k) are central factorial numbers of the second kind with odd indices. - Mircea Merca, Apr 04 2013
a(n) = (1+x^2)^(n+1/2) * (d/dx)^(2*n) (1+x^2)^(n-1/2).  See Tao link. - Robert Israel, Jun 04 2015
a(n) = 4^n * gamma(n + 1/2)^2 / Pi. - Daniel Suteu, Jan 06 2017
0 = a(n)*(+384*a(n+2) - 60*a(n+3) + a(n+4)) + a(n+1)*(-36*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) and a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 06 2017
From Robert FERREOL, Jul 30 2020: (Start)
a(n) = ((2*n)!/4^n)*binomial(2*n,n).
a(n) = (2*n-1)!*Sum_{k=0..n-1} a(k)/(2*k)!, n >= 1.
a(n) = A184877(2*n-1) for n>=1. (End)
From Amiram Eldar, Mar 18 2022: (Start)
Sum_{n>=0} 1/a(n) = 1 + L_0(1)*Pi/2, where L is the modified Struve function (see A197037).
Sum_{n>=0} (-1)^n/a(n) = 1 - H_0(1)*Pi/2, where H is the Struve function. (End)

Incorrect formula deleted by N. J. A. Sloane, Jul 03 2009

A111596 The matrix inverse of the unsigned Lah numbers A271703.

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 6, -6, 1, 0, -24, 36, -12, 1, 0, 120, -240, 120, -20, 1, 0, -720, 1800, -1200, 300, -30, 1, 0, 5040, -15120, 12600, -4200, 630, -42, 1, 0, -40320, 141120, -141120, 58800, -11760, 1176, -56, 1, 0, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016, -72, 1

Offset: 0

Wolfdieter Lang, Aug 23 2005

Also the associated Sheffer triangle to Sheffer triangle A111595.
Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-1,x), which equals (-1)^n * Lag(n,x,-1) below. Lag(n,Lag(.,x,-1),-1) = x^n evaluated umbrally, i.e., with (Lag(.,x,-1))^k = Lag(k,x,-1). - Tom Copeland, Apr 26 2014
Without row n=0 and column m=0 this is, up to signs, the Lah triangle A008297.
The unsigned column sequences are (with leading zeros): A000142, A001286, A001754, A001755, A001777, A001778, A111597-A111600 for m=1..10.
The row polynomials p(n,x) := Sum_{m=0..n} a(n,m)*x^m, together with the row polynomials s(n,x) of A111595 satisfy the exponential (or binomial) convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), n>=0.
Exponential Riordan array [1,x/(1+x)]. Inverse of the exponential Riordan array [1,x/(1-x)], which is the unsigned version of A111596. - Paul Barry, Apr 12 2007
For the unsigned subtriangle without column number m=0 and row number n=0, see A105278.
Unsigned triangle also matrix product |S1|*S2 of Stirling number matrices.
The unsigned row polynomials are Lag(n,-x,-1), the associated Laguerre polynomials of order -1 with negated argument. See Gradshteyn and Ryzhik, Abramowitz and Stegun and Rota (Finite Operator Calculus) for extensive formulas. - Tom Copeland, Nov 17 2007, Sep 09 2008
An infinitesimal matrix generator for unsigned A111596 is given by A132792. - Tom Copeland, Nov 22 2007
From the formalism of A132792 and A133314 for n > k, unsigned A111596(n,k) = a(k) * a(k+1)...a(n-1) / (n-k)! = a generalized factorial, where a(n) = A002378(n) = n-th term of first subdiagonal of unsigned A111596. Hence Deutsch's remark in A002378 provides an interpretation of A111596(n,k) in terms of combinations of certain circular binary words. - Tom Copeland, Nov 22 2007
Given T(n,k)= A111596(n,k) and matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 27 2008
Operationally, the unsigned row polynomials may be expressed as p_n(:xD:) = x*:Dx:^n*x^{-1}=x*D^nx^n*x^{-1}= n!*binomial(xD+n-1,n) = (-1)^n n! binomial(-xD,n) = n!L(n,-1,-:xD:), where, by definition, :AB:^n = A^nB^n for any two operators A and B, D = d/dx, and L(n,-1,x) is the Laguerre polynomial of order -1. A similarity transformation of the operators :Dx:^n generates the higher order Laguerre polynomials, which can also be expressed in terms of rising or falling factorials or Kummer's confluent hypergeometric functions (cf. the Mathoverflow post). -  Tom Copeland, Sep 21 2019

			Binomial convolution of row polynomials: p(3,x) = 6*x-6*x^2+x^3; p(2,x) = -2*x+x^2, p(1,x) = x, p(0,x) = 1,
together with those from A111595: s(3,x) = 9*x-6*x^2+x^3; s(2,x) = 1-2*x+x^2, s(1,x) = x, s(0,x) = 1; therefore
9*(x+y)-6*(x+y)^2+(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) = (6*y-6*y^2+y^3) + 3*x*(-2*y+y^2) + 3*(1-2*x+x^2)*y + 9*x-6*x^2+x^3.
From _Wolfdieter Lang_, Apr 28 2014: (Start)
The triangle a(n,m) begins:
n\m  0     1       2       3      4     5   6  7
0:   1
1:   0     1
2:   0    -2       1
3:   0     6      -6       1
4:   0   -24      36     -12      1
5:   0   120    -240     120    -20     1
6:   0  -720    1800   -1200    300   -30   1
7:   0  5040  -15120   12600  -4200   630 -42  1
...
For more rows see the link.
(End)

G. C. Greubel, Rows n=0..100 of triangle, flattened
Wolfdieter Lang, The first 11 rows of the triangle.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 4.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 6.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 20.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) 11.6.7.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers; Generators, Inversion, and Matrix, Binomial, and Integral Transforms; Lagrange a la Lah
A. Hennessy and P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Mathoverflow, Pochhammer symbol of a differential, and hypergeometric polynomials, a question posed by Emilio Pisanty and answered by Tom Copeland, 2012.
J. Taylor, Counting words with Laguerre polynomials, DMTCS Proc., Vol. AS, 2013, p. 1131-1142. [_Tom Copeland_, Jan 08 2016] [Broken link]
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96. [_Tom Copeland_, Dec 20 2018]
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Row sums: A111884. Unsigned row sums: A000262.
A002868 gives maximal element (in magnitude) in each row.
Cf. A130561 for a natural refinement.
Cf. A264428, A264429, A271703 (unsigned).
Cf. A008297, A089231, A105278 (variants).
Cf. A002378, A008277, A132792, A133314, A132792, A001263.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::odd, -(n+1)!, (n+1)!), 9); # Peter Luschny, Jan 27 2016

a[0, 0] = 1; a[n_, m_] := ((-1)^(n-m))*(n!/m!)*Binomial[n-1, m-1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)
T[ n_, k_] := (-1)^n n! Coefficient[ LaguerreL[ n, -1, x], x, k]; (* Michael Somos, Dec 15 2014 *)
rows = 9;
t = Table[(-1)^(n+1) n!, {n, 1, 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 *)

{T(n, k) = if( n<1 || k<1, n==0 && k==0, (-1)^n * n! * polcoeff( sum(k=1, n, binomial( n-1, k-1) * (-x)^k / k!), k))}; /* Michael Somos, Dec 15 2014 */

lah_number = lambda n, k: factorial(n-k)*binomial(n,n-k)*binomial(n-1,n-k)
A111596_row = lambda n: [(-1)^(n-k)*lah_number(n, k) for k in (0..n)]
for n in range(10): print(A111596_row(n)) # Peter Luschny, Oct 05 2014

# uses[inverse_bell_transform from A264429]
def A111596_matrix(dim):
    fact = [factorial(n) for n in (1..dim)]
    return inverse_bell_transform(dim, fact)
A111596_matrix(10) # Peter Luschny, Dec 20 2015

E.g.f. m-th column: ((x/(1+x))^m)/m!, m>=0.
E.g.f. for row polynomials p(n, x) is exp(x*y/(1+y)).
a(n, m) = ((-1)^(n-m))*|A008297(n, m)| = ((-1)^(n-m))*(n!/m!)*binomial(n-1, m-1), n>=m>=1; a(0, 0)=1; else 0.
a(n, m) = -(n-1+m)*a(n-1, m) + a(n-1, m-1), n>=m>=0, a(n, -1):=0, a(0, 0)=1; a(n, m)=0 if n|a(n,m)| = Sum_{k=m..n} |S1(n,k)|*S2(k,m), n>=0. S2(n,m):=A048993. S1(n,m):=A048994. - Wolfdieter Lang, May 04 2007
From Tom Copeland, Nov 21 2011: (Start)
For this Lah triangle, the n-th row polynomial is given umbrally by
  (-1)^n n! binomial(-Bell.(-x),n), where Bell_n(-x)= exp(x)(xd/dx)^n exp(-x), the n-th Bell / Touchard / exponential polynomial with neg. arg., (cf. A008277). E.g., 2! binomial(-Bell.(-x),2) = -Bell.(-x)*(-Bell.(-x)-1) = Bell_2(-x)+Bell_1(-x) = -2x+x^2.
A Dobinski relation is (-1)^n n! binomial(-Bell.(-x),n)= (-1)^n n! e^x Sum_{j>=0} (-1)^j binomial(-j,n)x^j/j!= n! e^x Sum_{j>=0} (-1)^j binomial(j-1+n,n)x^j/j!. See the Copeland link for the relation to inverse Mellin transform. (End)
The n-th row polynomial is (-1/x)^n e^x (x^2*D_x)^n e^(-x). - Tom Copeland, Oct 29 2012
Let f(.,x)^n = f(n,x) = x!/(x-n)!, the falling factorial,and r(.,x)^n = r(n,x) = (x-1+n)!/(x-1)!, the rising factorial, then the Lah polynomials, Lah(n,t)= n!*Sum{k=1..n} binomial(n-1,k-1)(-t)^k/k! (extra sign factor on odd rows), give the transform Lah(n,-f(.,x))= r(n,x), and Lah(n,r(.,x))= (-1)^n * f(n,x). - Tom Copeland, Oct 04 2014
|T(n,k)| = Sum_{j=0..2*(n-k)} A254881(n-k,j)*k^j/(n-k)!. Note that A254883 is constructed analogously from A254882. - Peter Luschny, Feb 10 2015
The T(n,k) are the inverse Bell transform of [1!,2!,3!,...] and |T(n,k)| are the Bell transform of [1!,2!,3!,...]. See A264428 for the definition of the Bell transform and A264429 for the definition of the inverse Bell transform. - Peter Luschny, Dec 20 2015
Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates a shifted, signed Narayana matrix A001263. - Tom Copeland, Sep 23 2020

New name using a comment from Wolfdieter Lang by Peter Luschny, May 10 2021

A111601 Exponential (binomial) convolution of A001818 (with interspersed zeros) and A000142 (factorials).

Original entry on oeis.org

1, 2, 9, 36, 225, 1350, 11025, 88200, 893025, 8930250, 108056025, 1296672300, 18261468225, 255660555150, 4108830350625, 65741285610000, 1187451971330625, 21374135483951250, 428670161650355625, 8573403233007112500

Offset: 1

Wolfdieter Lang, Aug 23 2005

Second column (m=1) of triangle |A111595|.

E.g.f.: (1/sqrt(1-x^2))*x/(1-x).
a(n) = n!*Sum_{k=0..floor((n-1)/2)} binomial(2*k, k)/(4^k).
a(2*k+1) = A111595(2*k+2, 0)= ((2*k+1)!!)^2 = A001818(k+1), k >= 0.
D-finite with recurrence (-n+1)*a(n) +n*a(n-1) +n*(n-1)^2*a(n-2)=0. a(1)=1, a(2)=2. - Sergei N. Gladkovskii, Jul 27 2012, corrected Aug 11 2025

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A111782 Ninth column (m=8) of unsigned triangle A111595.

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A111783 Tenth column (m=9) of unsigned triangle A111595.

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A001818 Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2.

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A111596 The matrix inverse of the unsigned Lah numbers A271703.

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A111601 Exponential (binomial) convolution of A001818 (with interspersed zeros) and A000142 (factorials).

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A001818 Squares of double factorials: (135*...(2n-1))^2 = ((2n-1)!!)^2.