A094310 -id:A094310 - cp's OEIS frontend

A165969 Triangle read by rows: T(n,m) = A094310(n,m)*A120070(n+1,m), 1 <= m <= n.

3, 16, 5, 90, 36, 14, 576, 252, 128, 54, 4200, 1920, 1080, 600, 264, 34560, 16200, 9600, 5940, 3456, 1560, 317520, 151200, 92400, 60480, 39312, 23520, 10800, 3225600, 1552320, 967680, 655200, 451584, 302400, 184320, 85680, 35925120, 17418240, 11007360, 7620480, 5443200, 3870720, 2643840, 1632960, 766080

Offset: 1

Paul Curtz, Oct 02 2009

The second array mentioned in the comment in A129326.

			Triangle begins
        3;
       16,       5;
       90,      36,     14;
      576,     252,    128,     54;
     4200,    1920,   1080,    600,    264;
    34560,   16200,   9600,   5940,   3456,   1560;
   317520,  151200,  92400,  60480,  39312,  23520,  10800;
  3225600, 1552320, 967680, 655200, 451584, 302400, 184320, 85680;

Cf. A077012, A079210.

A120070 := proc(n, m) n^2-m^2 ; end proc:
A094310 := proc(n,m) n!/m ; end proc:
A165969 := proc(n,m) A094310(n,m)*A120070(n+1,m) ; end proc: # R. J. Mathar, Jan 23 2011

A000254 Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1) = (n+1)*a(n) + n!.

Original entry on oeis.org

0, 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576, 10628640, 120543840, 1486442880, 19802759040, 283465647360, 4339163001600, 70734282393600, 1223405590579200, 22376988058521600, 431565146817638400, 8752948036761600000, 186244810780170240000

Offset: 0

N. J. A. Sloane

Number of permutations of n+1 elements with exactly two cycles.
Number of cycles in all permutations of [n]. Example: a(3) = 11 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132), (123) have 11 cycles altogether. - Emeric Deutsch, Aug 12 2004
Row sums of A094310: In the symmetric group S_n, each permutation factors into k independent cycles; a(n) = sum k over S_n. - Harley Flanders (harley(AT)umich.edu), Jun 28 2004
The sum of the top levels of the last column over all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, the levels of their last columns being 2 and 1, respectively. - Emeric Deutsch, Aug 12 2006
a(n) is divisible by n for all composite n >= 6. a(2*n) is divisible by 2*n + 1. - Leroy Quet, May 20 2007
For n >= 2 the determinant of the n-1 X n-1 matrix M(i,j) = i + 2 for i = j and 1 otherwise (i,j = 1..n-1). E.g., for n = 3 the determinant of [(3, 1), (1, 4)]. See 53rd Putnam Examination, 1992, Problem B5. - Franz Vrabec, Jan 13 2008, Mar 26 2008
The numerator of the fraction when we sum (without simplification) the terms in the harmonic sequence. (1 + 1/2 = 2/2 + 1/2 = 3/2; 3/2 + 1/3 = 9/6 + 2/6 = 11/6; 11/6 + 1/4 = 44/24 + 6/24 = 50/24;...). The denominator of this fraction is n!*A000142. - Eric Desbiaux, Jan 07 2009
The asymptotic expansion of the higher order exponential integral E(x,m=2,n=1) ~ exp(-x)/x^2*(1 - 3/x + 11/x^2 - 50/x^3 + 274/x^4 - 1764/x^5 + 13068/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009
a(n) is the number of permutations of [n+1] containing exactly 2 cycles. Example: a(2) = 3 because the permutations (1)(23), (12)(3), (13)(2) are the only permutations of [3] with exactly 2 cycles. - Tom Woodward (twoodward(AT)macalester.edu), Nov 12 2009
It appears that, with the exception of n= 4, a(n) mod n = 0 if n is composite and = n-1 if n is prime. - Gary Detlefs, Sep 11 2010
a(n) is a multiple of A025527(n). - Charles R Greathouse IV, Oct 16 2012
Numerator of harmonic number H(n) = Sum_{i=1..n} 1/i when not reduced. See A001008 (Wolstenholme numbers) for the reduced numerators. - Rahul Jha, Feb 18 2015
The Stirling transform of this sequence is A222058(n) (Harmonic-geometric numbers). - Anton Zakharov, Aug 07 2016
a(n) is the (n-1)-st elementary symmetric function of the first n numbers. - Anton Zakharov, Nov 02 2016
The n-th iterated integral of log(x) is x^n * (n! * log(x) - a(n))/(n!)^2 + a polynomial of degree n-1 with arbitrary coefficients. This can be proven using the recurrence relation a(n) = (n-1)! + n*a(n-1). - Mohsen Maesumi, Oct 31 2018
Primes p such that p^3 | a(p-1) are the Wolstenholme primes A088164. - Amiram Eldar and Thomas Ordowski, Aug 08 2019
Total number of left-to-right maxima (or minima) in all permutations of [n]. a(3) = 11 = 3+2+2+2+1+1: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21. - Alois P. Heinz, Aug 01 2020

			(1-x)^-1 * (-log(1-x)) = x + 3/2*x^2 + 11/6*x^3 + 25/12*x^4 + ...
G.f. = x + x^2 + 5*x^3 + 14*x^4 + 94*x^5 + 444*x^6 + 3828*x^7 + 25584*x^8 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, identities 186-190.
N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Dover Publications, 1986, see page 2. MR0863284 (89d:41049)
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 217.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.
Shanzhen Gao, Permutations with Restricted Structure (in preparation).
K. Javorszky, Natural Orders: De Ordinibus Naturalibus, 2016, ISBN 978-3-99057-139-2.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..449 (terms 0..100 from T. D. Noe)
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].
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159 (1996), 29-42.
J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, preprint, 2016.
FindStat - Combinatorial Statistic Finder, The number of cycles in the cycle decomposition of a permutation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 31.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv:1201.1323 [math.CO], 2012.
Sergey Kitaev and Jeffrey Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.
J. Riordan, Letter of 04/11/74.
John A. Rochowicz Jr., Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE), 8(2) (2015), Article 4.
N. A. Rosenberg, Informativeness of genetic markers for inference of ancestry, American Journal of Human Genetics 73 (2003), 1402-1422.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), #10.6.7, Section 4.3.2.
J. Scholes, 53rd Putnam 1992, Problem B5.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles. (Annotated scans of some selected pages)
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 5.

Cf. A000399, A000454, A000482, A001233, A001234, A243569, A243570.
Cf. A000774, A004041, A024167, A046674, A049034, A008275.
Cf. A021009, A081358, A092691, A094587, A151881, A121633.
With signs: A081048.
Column 1 in triangle A008969.
Row sums of A136662.

a:=[]; for n in [1..22] do a:=a cat [Abs(StirlingFirst(n,2))]; end for; a; // Marius A. Burtea, Jan 01 2020

A000254 := proc(n) option remember; if n<=1 then n else n*A000254(n-1)+(n-1)!; fi; end: seq(A000254(n),n=0..21);
a := n -> add(n!/k, k=1..n): seq(a(n), n=0..21); # Zerinvary Lajos, Jan 22 2008

Table[ (PolyGamma[ m ]+EulerGamma) (m-1)!, {m, 1, 24} ] (* Wouter Meeussen *)
Table[ n!*HarmonicNumber[n], {n, 0, 19}] (* Robert G. Wilson v, May 21 2005 *)
Table[Sum[1/i,{i,1,n}]/Product[1/i,{i,1,n}],{n,1,30}] (* Alexander Adamchuk, Jul 11 2006 *)
Abs[StirlingS1[Range[20],2]] (* Harvey P. Dale, Aug 16 2011 *)
Table[Gamma'[n + 1] /. EulerGamma -> 0, {n, 0, 30}] (* Li Han, Feb 14 2024*)

a(n):=(-1)^(n+1)/2*(n+1)*sum(k*bern(k-1)*stirling1(n,k),k,1,n); /* Vladimir Kruchinin, Nov 20 2016 */

A000254 := proc(n) begin n*A000254(n-1)+fact(n-1) end_proc: A000254(1) := 1:

{a(n) = if( n<0, 0, (n+1)! / 2 * sum( k=1, n, 1 / k / (n+1-k)))} /* Michael Somos, Feb 05 2004 */

[stirling_number1(i, 2) for i in range(1, 22)]  # Zerinvary Lajos, Jun 27 2008

Let P(n,X) = (X+1)*(X+2)*(X+3)*...*(X+n); then a(n) is the coefficient of X; or a(n) = P'(n,0). - Benoit Cloitre, May 09 2002
Sum_{k > 0} a(k) * x^k/ k!^2 = exp(x) *(Sum_{k>0} (-1)^(k+1) * x^k / (k * k!)). - Michael Somos, Mar 24 2004; corrected by Warren D. Smith, Feb 12 2006
a(n) is the coefficient of x^(n+2) in (-log(1-x))^2, multiplied by (n+2)!/2.
a(n) = n! * Sum_{i=1..n} 1/i = n! * H(n), where H(n) = A001008(n)/A002805(n) is the n-th harmonic number.
a(n) ~ 2^(1/2)*Pi^(1/2)*log(n)*n^(1/2)*e^-n*n^n. - Joe Keane (jgk(AT)jgk.org), Jun 06 2002
E.g.f.: log(1 - x) / (x-1). (= (log(1 - x))^2 / 2 if offset 1). - Michael Somos, Feb 05 2004
D-finite with recurrence: a(n) = a(n-1) * (2*n - 1) - a(n-2) * (n - 1)^2, if n > 1. - Michael Somos, Mar 24 2004
a(n) = A081358(n)+A092691(n). - Emeric Deutsch, Aug 12 2004
a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k. - Vladeta Jovovic, Jan 29 2005
p^2 divides a(p-1) for prime p > 3. a(n) = (Sum_{i=1..n} 1/i) / Product_{i=1..n} 1/i. - Alexander Adamchuk, Jul 11 2006
a(n) = 3* A001710(n) + 2* A001711(n-3) for n > 2; e.g., 11 = 3*3 + 2*1, 50 = 3*12 + 2*7, 274 = 3*60 + 2*47, ... - Gary Detlefs, May 24 2010
a(n) = A138772(n+1) - A159324(n). - Gary Detlefs, Jul 05 2010
a(n) = A121633(n) + A002672(n). - Gary Detlefs, Jul 18 2010
a(n+1) = Sum_{i=1..floor((n-1)/2)} n!/((n-i)*i) + Sum_{i=ceiling(n/2)..floor(n/2)} n!/(2*(n-i)*i). - Shanzhen Gao, Sep 14 2010
From Gary Detlefs, Sep 11 2010: (Start)
a(n) = (a(n-1)*(n^2 - 2*n + 1) + (n + 1)!)/(n - 1) for n > 2.
It appears that, with the exception of n = 2, (a(n+1)^2 - a(n)^2) mod n^2 = 0 if n is composite and 4*n if n is prime.
It appears that, with the exception of n = 2, (a(n+1)^3 - a(n)^2) mod n = 0 if n is composite and n - 2 if n is prime.
It appears that, with the exception of n = 2, (a(n)^2 + a(n+1)^2) mod n = 0 if n is composite and = 2 if n is prime. (End)
a(n) = Integral_{x=0..oo} (x^n - n!)*log(x)*exp(-x) dx. - Groux Roland, Mar 28 2011
a(n) = 3*n!/2 + 2*(n-2)!*Sum_{k=0..n-3} binomial(k+2,2)/(n-2-k) for n >= 2. - Gary Detlefs, Sep 02 2011
a(n)/(n-1)! = ml(n) = n*ml(n-1)/(n-1) + 1 for n > 1, where ml(n) is the average number of random draws from an n-set with replacement until the total set has been observed. G.f. of ml: x*(1 - log(1 - x))/(1 - x)^2. - Paul Weisenhorn, Nov 18 2011
a(n) = det(|S(i+2, j+1)|, 1 <= i,j <= n-2), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
E.g.f.: x/(1 - x)*E(0)/2, where E(k) = 2 + E(k+1)*x*(k + 1)/(k + 2). - Sergei N. Gladkovskii, Jun 01 2013 [Edited by Michael Somos, Nov 28 2013]
0 = a(n) * (a(n+4) - 6*a(n+3) + 7*a(n+2) - a(n+1)) - a(n+1) * (4*a(n+3) - 6*a(n+2) + a(n+1)) + 3*a(n+2)^2 unless n=0. - Michael Somos, Nov 28 2013
For a simple way to calculate the sequence, multiply n! by the integral from 0 to 1 of (1 - x^n)/(1 - x) dx. - Rahul Jha, Feb 18 2015
From Ilya Gutkovskiy, Aug 07 2016: (Start)
Inverse binomial transform of A073596.
a(n) ~ sqrt(2*Pi*n) * n^n * (log(n) + gamma)/exp(n), where gamma is the Euler-Mascheroni constant A001620. (End)
a(n) = ((-1)^(n+1)/2*(n+1))*Sum_{k=1..n} k*Bernoulli(k-1)*Stirling1(n,k). - Vladimir Kruchinin, Nov 20 2016
a(n) = (n)! * (digamma(n+1) + gamma), where gamma is the Euler-Mascheroni constant A001620. - Pedro Caceres, Mar 10 2018
From Andy Nicol, Oct 21 2021: (Start)
Gamma'(x) = a(x-1) - (x-1)!*gamma, where Gamma'(x) is the derivative of the gamma function at positive integers and gamma is the Euler-Mascheroni constant. E.g.:
Gamma'(1) = -gamma, Gamma'(2) = 1-gamma, Gamma'(3) = 3-2*gamma,
Gamma'(22) = 186244810780170240000 - 51090942171709440000*gamma. (End)
From Peter Bala, Feb 03 2022: (Start)
The following are all conjectural:
E.g.f.: for nonzero m, (1/m)*Sum_{n >= 1} (-1)^(n+1)*(1/n)*binomial(m*n,n)* x^n/(1 - x)^(m*n+1) = x + 3*x^2/2! + 11*x^3/3! + 50*x^4/4! + ....
For nonzero m, a(n) = (1/m)*n!*Sum_{k = 1..n} (-1)^(k+1)*(1/k)*binomial(m*k,k)* binomial(n+(m-1)*k,n-k).
a(n)^2 = (1/2)*n!^2*Sum_{k = 1..n} (-1)^(k+1)*(1/k^2)*binomial(n,k)* binomial(n+k,k). (End)
From Mélika Tebni, Jun 20 2022: (Start)
a(n) = -Sum_{k=0..n} k!*A021009(n, k+1).
a(n) = Sum_{k=0..n} k!*A094587(n, k+1). (End)
a(n) = n! * 1/(1 - 1^2/(3 - 2^2/(5 - 3^2/(7 - ... - (n - 1)^2/((2*n - 1)))))). - Peter Bala, Mar 16 2024

A110468 a(n) = (2*n + 1)!/(n + 1).

Original entry on oeis.org

1, 3, 40, 1260, 72576, 6652800, 889574400, 163459296000, 39520825344000, 12164510040883200, 4644631106519040000, 2154334728240414720000, 1193170003333152768000000, 777776389315596582912000000, 589450799582646796969574400000, 513927415886120176107847680000000

Offset: 0

Paul Barry, Jul 21 2005

Convolution of (-1)^n*n! and n! with interpolated zeros suppressed.
Denominator of absolute value of coefficient of 1/(x+n^2) in the partial fraction decomposition of 1/(x+1)*1/(x+4)*..*1/(x+n^2). - Joris Roos (jorisr(AT)gmx.de), Aug 07 2009
With offset = 1: a(n) is the number of permutations of {1,2,...,2n} composed of two cycles of length n. - Geoffrey Critzer, Nov 11 2012

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A094310, A143623, A145877, A202768, A346085.

Table[(2n)!/(2n^2),{n,1,20}] (* Geoffrey Critzer, Nov 11 2012 *)

for(n=0,50, print1((2*n+1)!/(n+1), ", ")) \\ G. C. Greubel, Aug 28 2017

E.g.f.: log((1-x)*(1+x))/(-x).
a(n) = (2*n)!*Sum_{k = 0..2*n} (-1)^k/binomial(2*n, k).
a(n) = Sum_{k = 0..2*n} k!*(-1)^k*(2*n-k)!.
Sum_{n>=0} 1/a(n) = e/2. - Franz Vrabec, Jan 17 2008
(n+1)*a(n) + 2*(-n^2)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Nov 15 2012
a(n) = Product_{i=1..n} (n+1-i)*(n+1+i). - Vaclav Kotesovec, Oct 21 2014
a(n) = A145877(2*n+2, n+1). - Alois P. Heinz, Apr 21 2017
a(n) = A346085(2*n+2, n+1). - Alois P. Heinz, Jul 04 2021
Sum_{n>=0} (-1)^n/a(n) = (cos(1) + sin(1))/2 = (1/2) * A143623. - Amiram Eldar, Feb 08 2022
a(p-1) == 1 (mod p), p a prime. - Peter Bala, Jul 29 2024
Sum_{n>=0} x^(2*n+1)/a(n) = (sinh(x) + x*cosh(x))/2. - Michael Somos, Jul 23 2025

Simpler definition from Robert Israel, Jul 20 2006

A129825 a(n) = n!*Bernoulli(n-1), n > 2; a(0)=0, a(1)=1, a(2)=1.

Original entry on oeis.org

0, 1, 1, 1, 0, -4, 0, 120, 0, -12096, 0, 3024000, 0, -1576143360, 0, 1525620096000, 0, -2522591034163200, 0, 6686974460694528000, 0, -27033456071346536448000, 0, 160078872315904478576640000, 0, -1342964491649083924630732800000, 0, 15522270327163593186886877184000000, 0

Offset: 0

Paul Curtz, Jun 03 2007

sign

Define "conjugated" Bernoulli numbers G(n) via G(0)=0, G(1)=B(0)=1, G(2)=-B(1)=1/2, G(n+1)=B(n), where B(n)=A027641(n)/A027642(n).
The sequence is then defined by a(n) = n!*G(n).
The first differences are 1, 0, 0, -1, -4, 4, 120, -120, -12096, ...
The 2nd differences are -1, 0, -1, -3, 8, 116, -240, -11976, 24192, 3011904, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A001332, A027641, A027642, A129716, A129814, A129826.
Equals second left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A161742 and A161743.
Cf. A094310 [T(n,k) = n!/k], A008277 [S2(n,k); Stirling numbers of the second kind], A028246 [Worpitzky's triangle] and A008955 [CFN triangle].

[n le 2 select Floor((n+1)/2) else Factorial(n)*Bernoulli(n-1): n in [0..40]]; // G. C. Greubel, Apr 26 2024

A129825 := proc(n) if n <= 1 then n; elif n = 2 then 1; else n!*bernoulli(n-1) ; fi; end: # R. J. Mathar, May 21 2009

a[n_] := n!*BernoulliB[n-1]; a[0]=0; a[2]=1; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Mar 04 2013 *)

[(n+1)//2 if n <3 else factorial(n)*bernoulli(n-1) for n in range(41)] # G. C. Greubel, Apr 26 2024

From Johannes W. Meijer, Jun 18 2009: (Start)
a(n) = Sum_{k=1..n} (-1)^(k+1)*(n!/k)*S2(n, k)*(k-1)!.
a(n) = Sum_{k=0..n-1} ((-1)^k/(k!*(k+1)!))*n!*A028246(n, k+1) *A008955(k, k). (End)
a(n) = A129814(n-1) for n > 2. - Georg Fischer, Oct 07 2018

Edited by R. J. Mathar, May 21 2009

A166350 Triangle read by rows: T(n,m) = m!, n >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 6, 24, 1, 2, 6, 24, 120, 1, 2, 6, 24, 120, 720, 1, 2, 6, 24, 120, 720, 5040, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 1, 2, 6, 24, 120, 720, 5040, 40320

Offset: 1

Paul Curtz, Oct 12 2009

			Triangle begins:
  1;
  1, 2;
  1, 2, 6;
  1, 2, 6, 24;
  1, 2, 6, 24, 120;
  1, 2, 6, 24, 120, 720;
  1, 2, 6, 24, 120, 720, 5040;
  ...

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
Index entries for sequences related to factorial numbers

Cf. A000142, A002260, A094310, A077012, A079210.
Cf. A014454.
Row sums give A007489.

import Data.List (inits)
a166350 n k = a166350_tabl !! (n-1) !! (n-1)
a166350_row n = a166350_tabl !! (n-1)
a166350_tabl = tail $ inits $ tail a000142_list
-- Reinhard Zumkeller, Nov 11 2013

Flatten[Table[Range[n]!,{n,11}]] (* Harvey P. Dale, Jan 06 2012 *)
Module[{nn=20,fs},fs=Range[nn]!;Table[Take[fs,n],{n,nn}]]//Flatten (* Harvey P. Dale, Jun 14 2020 *)

T(n,m) = A000142(m).

Definition clarified - R. J. Mathar, Oct 14 2009

A203239 Odd numbered terms of the sequence whose n-th term is the (n-1)-st elementary symmetric function of (i, 2i, 3i, ..., ni), where i=sqrt(-1).

Original entry on oeis.org

3, -50, 1764, -109584, 10628640, -1486442880, 283465647360, -70734282393600, 22376988058521600, -8752948036761600000, 4148476779335454720000, -2342787216398718566400000, 1554454559147562279567360000

Offset: 1

Clark Kimberling, Dec 30 2011

sign

			The first 10 terms of the "full sequence" are as follows:
1, 3i, -11, -50i, 274, 1764i, -13068, -109584i, 1026576, 10628640i;
Abbreviate "elementary symmetric function" as esf. Then, starting with {i, 2i, 3i, 4i, ...}:
0th esf of {i}: 1
1st esf of {i, 2i}: i+2i = 3i
2nd esf of {i, 2i, 3i}: -2-3-6 = -11.
For the alternating terms 3i, -50i, ..., see A203240.

Cf. A000254, A094310, A165675, A203240.

f[k_] := k*I; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}]
Table[-I*a[2 n], {n, 1, 22}]     (* A203239 *)
Table[a[2 n - 1], {n, 1, 22}]    (* A203240 *)
Table[(-1)^(n + 1)*(2*n)!*HarmonicNumber[2*n], {n, 13}] (* Arkadiusz Wesolowski, Mar 25 2013 *)

a(n) = (-1)^(n+1)*(2*n)!*Sum_{i=1..2n} 1/i. - Arkadiusz Wesolowski, Mar 25 2013
From Anton Zakharov, Oct 26 2016: (Start)
a(n) = (-1)^(n+1)*Sum_{k=1..n} A094310(2n,k).
(-1)^(n+1)*a(n) = A000254(2n) (signed bisection of A000254). (End)

A203240 Real part of even numbered terms of the sequence s(n)=(n-1)-st elementary symmetric function of (i, 2i, 3i,...,ni).

Original entry on oeis.org

1, -11, 274, -13068, 1026576, -120543840, 19802759040, -4339163001600, 1223405590579200, -431565146817638400, 186244810780170240000, -96538966652493066240000, 59190128811701203599360000, -42373564558110787183902720000

Offset: 1

Clark Kimberling, Dec 30 2011

sign

			See A203239.

Cf. A203239, A094310, A000254.

a := n -> (-1)^(n-1)*(2*n-1)!*harmonic(2*n-1):
seq(a(n), n = 1..14); # Peter Luschny, Oct 26 2016

f[k_] := k*I; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 22}]
Table[-I*a[2 n], {n, 1, 22}]     (* A203239 *)
Table[a[2 n - 1], {n, 1, 22}]    (* A203240 *)
Table[(-1)^(n + 1)*(2*n - 1)!*HarmonicNumber[2*n - 1], {n, 14}] (* Arkadiusz Wesolowski, Mar 25 2013 *)

a(n) = (-1)^(n+1)*(2*n - 1)!*Sum(i=1..2*n-1, 1/i). - Arkadiusz Wesolowski, Mar 25 2013
From Anton Zakharov, Oct 26 2016: (Start)
a(n) = (-1)^(n+1)*Sum_{k=1..n} A094310(2n-1,k).
(-1)^(n+1)*a(n) = A000254(2*n-1) (signed bisection of A000254). (End)

A120435 Triangle read by rows: T(n,k) = lcm(1,2,3,4,...,n)/k (1 <= k <= n).

Original entry on oeis.org

1, 2, 1, 6, 3, 2, 12, 6, 4, 3, 60, 30, 20, 15, 12, 60, 30, 20, 15, 12, 10, 420, 210, 140, 105, 84, 70, 60, 840, 420, 280, 210, 168, 140, 120, 105, 2520, 1260, 840, 630, 504, 420, 360, 315, 280, 2520, 1260, 840, 630, 504, 420, 360, 315, 280, 252, 27720, 13860, 9240

Offset: 1

Leroy Quet, Jul 15 2006

T(n,1) = A003418(n). Row sums yield A025529. - Emeric Deutsch, Jul 24 2006

			Triangle starts:
   1;
   2,  1;
   6,  3,  2;
  12,  6,  4,  3;
  60, 30, 20, 15, 12;

John Tyler Rascoe, Rows n = 1..140, flattened

Cf. A003418, A025529, A094310.

T:=proc(n,k) if k<=n then lcm(seq(j,j=1..n))/k else 0 fi end: for n from 1 to 11 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form); # Emeric Deutsch, Jul 24 2006

from math import lcm
def A120435(maxrow):
    A,rlcm = [],1
    for n in range(1,maxrow+1):
        rlcm = lcm(n,rlcm)
        A.append(list(rlcm//k for k in range(1,n+1)))
    return A # John Tyler Rascoe, Nov 08 2024

More terms from Emeric Deutsch, Jul 24 2006

cp's OEIS Frontend

A165969 Triangle read by rows: T(n,m) = A094310(n,m)*A120070(n+1,m), 1 <= m <= n.

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A000254 Unsigned Stirling numbers of first kind, s(n+1,2): a(n+1) = (n+1)*a(n) + n!.

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A129825 a(n) = n!*Bernoulli(n-1), n > 2; a(0)=0, a(1)=1, a(2)=1.

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A166350 Triangle read by rows: T(n,m) = m!, n >= 1.

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A203239 Odd numbered terms of the sequence whose n-th term is the (n-1)-st elementary symmetric function of (i, 2i, 3i, ..., ni), where i=sqrt(-1).

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A203240 Real part of even numbered terms of the sequence s(n)=(n-1)-st elementary symmetric function of (i, 2i, 3i,...,ni).

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