A099285 -id:A099285 - cp's OEIS frontend

A163931 Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1.

0, 9, 7, 8, 4, 3, 1, 9, 7, 2, 1, 6, 6, 7, 0, 1, 7, 9, 3, 2, 5, 5, 3, 7, 7, 8, 9, 0, 4, 5, 2, 8, 0, 0, 8, 2, 7, 6, 9, 5, 8, 2, 2, 6, 9, 5, 3, 0, 2, 6, 5, 7, 6, 5, 5, 7, 4, 4, 2, 1, 2, 4, 2, 4, 5, 4, 4, 7, 1, 3, 7, 6, 2, 6, 1, 4, 0, 9, 0, 4, 8, 8, 7, 3, 6, 9, 6, 0, 4, 8, 9, 1, 8, 5, 5, 5, 0, 8, 9, 4, 5, 4, 6, 7, 0

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

We define the higher-order exponential integrals by E(x,m,n) = x^(n-1)*Integral_{t=x..infinity} E(t,m-1,n)/t^n for m >= 1 and n >= 1 with E(x,m=0,n) = exp(-x), see Meijer and Baken.
The properties of the E(x,m,n) are analogous to those of the well-known exponential integrals E(x,m=1,n), see Abramowitz and Stegun and the formulas.
The series expansions of the higher-order exponential integrals are dominated by the constants alpha(k,n), see A163927, and gamma(k,n) = G(k,n), see A090998.
For information about the asymptotic expansion of the E(x,m,n) see A163932.
Values of E(x,m,n) can be evaluated with the Maple program.

			E(1,2,1) = 0.09784319721667017932553778904528008276958226953026576557442124245....

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251.
J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.
M. S. Milgram, The generalized integro-exponential function, Math. of Computation, Vol. 44, pp. 443-458, 1985.
Eric Weisstein's World of Mathematics, The Exponential Integral.

Cf. A163927 (alpha(k,n)), A090998 (gamma(k,n) = G(k,n)), A163932.
Cf. A068985 (E(x=1,m=0,n) = exp(-1)) and A099285 (E(x=1,m=1,n=1)).
Cf. A001563 (n*n!), A002775 (n^2*n!), A091363 (n^3*n!) and A091364 (n^4*n!).

E:= proc(x,m,n) local nmax, kmax, EI, k1, k2, n1, n2; option remember: nmax:=20; kmax:=20; k1:=0: for n1 from 0 to nmax do alpha(k1,n1):=1 od: for k1 from 1 to kmax do for n1 from 1 to nmax do alpha(k1,n1) := (1/k1)*sum(sum(p^(-2*(k1-i1)),p=0..n1-1)*alpha(i1, n1),i1=0..k1-1) od; od: for n2 from 0 to kmax do G(0,n2):=1 od: for n2 from 1 to nmax do for k2 from 1 to kmax do G(k2,n2):=(1/k2)*(((gamma-sum(p^(-1),p=1..n2-1))*G(k2-1,n2)+ sum((Zeta(k2-i2)-sum(p^(-(k2-i2)), p=1..n2-1))*G(i2,n2),i2=0..k2-2))) od; od: EI:= evalf((-1)^m*((-x)^(n-1)/(n-1)!*sum(alpha(kz,n)*(G(m-2*kz,n)+sum(G(m-2*kz-i,n)*ln(x)^i/i!,i=1..m-2*kz)), kz=0..floor(m/2)) + sum((-x)^kx/((kx-n+1)^m*kx!),kx=0..n-2) + sum((-x)^ky/((ky-n+1)^m*ky!),ky=n..infinity))); return(EI): end:

Join[{0}, RealDigits[ N[ EulerGamma^2/2 + Pi^2/12 - HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -1], 104]][[1]]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *)

t=1; Euler^2/2 + Pi^2/12 + sumalt(k=1, t*=k; (-1)^k/(k^2*t)) \\ Charles R Greathouse IV, Nov 07 2016

E(x=1,m=2,n=1) = gamma^2/2 + Pi^2/12 + Sum_{k>=1} ((-1)^k/(k^2*k!)).
E(x=0,n,m) = (1/(n-1))^m for n >= 2.
Integral_{t=0..x} E(t,m,n) = 1/n^m - E(x,n,n+1).
dE(x,m,n+1)/dx = - E(x,m,n).
E(x,m,n+1) = (1/n)*(E(x,m-1,n+1) - x*E(x,m,n)).
E(x,m,n) = (-1)^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)}(alpha (kz, n)*G(m-2*kz, n)) + (-1) ^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)}(Sum_{i=1..m-2*kz}(alpha (kz, n) *G(m-2*kz-i, n)*log(x)^i/i!)) + (-1)^m * Sum_{ kx=0..n-2}((-x)^kx/((kx-n+1)^m*kx!) + (-1)^m * Sum_{ky>=n}((-x)^ky /(( ky-n+1)^m*ky!)).

A001286 Lah numbers: a(n) = (n-1)*n!/2.

Original entry on oeis.org

1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600, 199584000, 2634508800, 37362124800, 566658892800, 9153720576000, 156920924160000, 2845499424768000, 54420176498688000, 1094805903679488000, 23112569077678080000, 510909421717094400000

Offset: 2

N. J. A. Sloane

Number of surjections from {1,...,n} to {1,...,n-1}. - Benoit Cloitre, Dec 05 2003
First Eulerian transform of 0,1,2,3,4,... . - Ross La Haye, Mar 05 2005
With offset 0 : determinant of the n X n matrix m(i,j)=(i+j+1)!/i!/j!. - Benoit Cloitre, Apr 11 2005
These numbers arise when expressing n(n+1)(n+2)...(n+k)[n+(n+1)+(n+2)+...+(n+k)] as sums of squares: n(n+1)[n+(n+1)] = 6(1+4+9+16+ ... + n^2), n(n+1)(n+2)(n+(n+1)+(n+2)) = 36(1+(1+4)+(1+4+9)+...+(1+4+9+16+ ... + n^2)), n(n+1)(n+2)(n+3)(n+(n+1)+(n+2)+(n+3)) = 240(...), ... . - Alexander R. Povolotsky, Oct 16 2006
a(n) is the number of edges in the Hasse diagram for the weak Bruhat order on the symmetric group S_n. For permutations p,q in S_n, q covers p in the weak Bruhat order if p,q differ by an adjacent transposition and q has one more inversion than p. Thus 23514 covers 23154 due to the transposition that interchanges the third and fourth entries. Cf. A002538 for the strong Bruhat order. - David Callan, Nov 29 2007
a(n) is also the number of excedances in all permutations of {1,2,...,n} (an excedance of a permutation p is a value j such p(j)>j). Proof: j is exceeded (n-1)! times by each of the numbers j+1, j+2, ..., n; now, Sum_{j=1..n} (n-j)(n-1)! = n!(n-1)/2. Example: a(3)=6 because the number of excedances of the permutations 123, 132, 312, 213, 231, 321 are 0, 1, 1, 1, 2, 1, respectively. - Emeric Deutsch, Dec 15 2008
(-1)^(n+1)*a(n) is the determinant of the n X n matrix whose (i,j)-th element is 0 for i = j, is j-1 for j>i, and j for j < i. - Michel Lagneau, May 04 2010
Row sums of the triangle in A030298. - Reinhard Zumkeller, Mar 29 2012
a(n) is the total number of ascents (descents) over all n-permutations. a(n) = Sum_{k=1..n} A008292(n,k)*k. - Geoffrey Critzer, Jan 06 2013
For m>=4, a(m-2) is the number of Hamiltonian cycles in a simple graph with m vertices which is complete, except for one edge. Proof: think of distinct round-table seatings of m persons such that persons "1" and "2" may not be neighbors; the count is (m-3)(m-2)!/2. See also A001710. - Stanislav Sykora, Jun 17 2014
Popularity of left (right) children in treeshelves. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n. See A278677, A278678 or A278679 for more definitions and examples. See A008292 for the distribution of the left (right) children in treeshelves. - Sergey Kirgizov, Dec 24 2016

			G.f. = x^2 + 6*x^3 + 36*x^4 + 240*x^5 + 1800*x^6 + 15120*x^7 + 141120*x^8 + ...
a(10) = (1+2+3+4+5+6+7+8+9)*(1*2*3*4*5*6*7*8*9) = 16329600. - _Reinhard Zumkeller_, May 15 2010

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 90, ex. 4.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
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 = 2..100
Yasmin Aguillon et al., On Parking Functions and The Tower of Hanoi, arXiv:2206.00541 [math.CO], 2022.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Jeremy L. Martin, Amanda Priestley, and Gabe Udell, Statistics on L-interval parking functions, arXiv:2507.07243 [math.CO], 2025. See p. 7.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 399.
Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Martínez Mori, Boolean intervals in the weak order of S_n, arXiv:2306.14734 [math.CO], 2023.
Lucas Chaves Meyles, Pamela E. Harris, Richter Jordaan, Gordon Rojas Kirby, Sam Sehayek, and Ethan Spingarn, Unit-Interval Parking Functions and the Permutohedron, arXiv:2305.15554 [math.CO], 2023.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Sandi Klavžar, Uroš Milutinović and Ciril Petr, Hanoi graphs and some classical numbers, Expo. Math. 23 (2005), no. 4, 371-378.
Siegfried Lehr, Jeffrey Shallit, and John Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Permutation Ascent.

Cf. A001710, A052609, A062119, A075181, A060638, A060608, A060570, A060612, A135218, A019538, A053495, A051683, A213168, A278677, A278678, A278679, A008292.
A002868 is an essentially identical sequence.
Column 2 of |A008297|.
Third column (m=2) of triangle |A111596(n, m)|: matrix product of |S1|.S2 Stirling number matrices.
Cf. also A000110, A000111.
Cf. A001113, A001620, A091725, A099285.

a001286 n = sum[1..n-1] * product [1..n-1]
-- Reinhard Zumkeller, Aug 01 2011

[(n-1)*Factorial(n)/2: n in [2..25]]; // Vincenzo Librandi, Sep 09 2016

seq(sum(mul(j,j=3..n), k=2..n), n=2..21); # Zerinvary Lajos, Jun 01 2007

Table[Sum[n!, {i, 2, n}]/2, {n, 2, 20}] (* Zerinvary Lajos, Jul 12 2009 *)
nn=20;With[{a=Accumulate[Range[nn]],t=Range[nn]!},Times@@@Thread[{a,t}]] (* Harvey P. Dale, Jan 26 2013 *)
Table[(n - 1) n! / 2, {n, 2, 30}] (* Vincenzo Librandi, Sep 09 2016 *)

A001286(n):=(n-1)*n!/2$
makelist(A001286(n),n,1,30); /* Martin Ettl, Nov 03 2012 */

a(n)=(n-1)*n!/2 \\ Charles R Greathouse IV, Nov 20 2012

from _future_ import division
A001286_list = [1]
for n in range(2,100):
    A001286_list.append(A001286_list[-1]*n*(n+1)//(n-1)) # Chai Wah Wu, Apr 11 2018

[(n-1)*factorial(n)/2 for n in range(2, 21)] # Zerinvary Lajos, May 16 2009

a(n) = Sum_{i=0..n-1} (-1)^(n-i-1) * i^n * binomial(n-1,i). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000 [corrected by Amiram Eldar, May 02 2022]
E.g.f.: x^2/[2(1-x)^2]. - Ralf Stephan, Apr 02 2004
a(n+1) = (-1)^(n+1)*det(M_n) where M_n is the n X n matrix M_(i,j)=max(i*(i+1)/2,j*(j+1)/2). - Benoit Cloitre, Apr 03 2004
Row sums of table A051683. - Alford Arnold, Sep 29 2006
5th binomial transform of A135218: (1, 1, 1, 25, 25, 745, 3145, ...). - Gary W. Adamson, Nov 23 2007
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n)=(-1)^n*f(n,2,-2), (n>=2). - Milan Janjic, Mar 01 2009
a(n) = A000217(n-1)*A000142(n-1). - Reinhard Zumkeller, May 15 2010
a(n) = (n+1)!*Sum_{k=1..n-1} 1/(k^2+3*k+2). - Gary Detlefs, Sep 14 2011
Sum_{n>=2} 1/a(n) = 2*(2 - exp(1) - gamma + Ei(1)) = 1.19924064599..., where gamma = A001620 and Ei(1) = A091725. - Ilya Gutkovskiy, Nov 24 2016
a(n+1) = a(n)*n*(n+1)/(n-1). - Chai Wah Wu, Apr 11 2018
Sum_{n>=2} (-1)^n/a(n) = 2*(gamma - Ei(-1)) - 2/e, where e = A001113 and Ei(-1) = -A099285. - Amiram Eldar, May 02 2022

A073003 Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant, or the Euler-Gompertz constant.

Original entry on oeis.org

5, 9, 6, 3, 4, 7, 3, 6, 2, 3, 2, 3, 1, 9, 4, 0, 7, 4, 3, 4, 1, 0, 7, 8, 4, 9, 9, 3, 6, 9, 2, 7, 9, 3, 7, 6, 0, 7, 4, 1, 7, 7, 8, 6, 0, 1, 5, 2, 5, 4, 8, 7, 8, 1, 5, 7, 3, 4, 8, 4, 9, 1, 0, 4, 8, 2, 3, 2, 7, 2, 1, 9, 1, 1, 4, 8, 7, 4, 4, 1, 7, 4, 7, 0, 4, 3, 0, 4, 9, 7, 0, 9, 3, 6, 1, 2, 7, 6, 0, 3, 4, 4, 2, 3, 7

Offset: 0

Robert G. Wilson v, Aug 03 2002

0! - 1! + 2! - 3! + 4! - 5! + ... = (Borel) Sum_{n>=0} (-y)^n n! = KummerU(1,1,1/y)/y.
Decimal expansion of phi(1) where phi(x) = Integral_{t>=0} e^-t/(x+t) dt. - Benoit Cloitre, Apr 11 2003
The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m => -1, is intimately related to Gompertz's constant. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003) with A000110 the Bell numbers and A040027 a sequence that was published by Gould, see for more information A163940. - Johannes W. Meijer, Oct 16 2009
Named by Le Lionnais (1983) after the English self-educated mathematician and actuary Benjamin Gompertz (1779 - 1865). It was named the Euler-Gompertz constant by Finch (2003). Lagarias (2013) noted that he has not located this constant in Gompertz's writings. - Amiram Eldar, Aug 15 2020

			0.59634736232319407434107849936927937607417786015254878157348491...
With n := 10^5, Sum_{k >= 0} (n/(n + 1))^k/(n + k) = 0.5963(51...). - _Peter Bala_, Jun 19 2024

Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 171
Bruce C. Berndt, Ramanujan's notebooks Part I, Springer, p. 144-145.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 303, 424-425.
Francois Le Lionnais, Les nombres remarquables, Paris: Hermann, 1983. See p. 29.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 44, page 426.
H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948, p. 356.

Robert Price, Table of n, a(n) for n = 0..10000
A. I. Aptekarev, On linear forms containing the Euler constant, arXiv:0902.1768 [math.NT], 2009. - _R. J. Mathar_, Feb 14 2009
Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.
Xiangyu Ding and Lisa Hui Sun, Proof of Merca's stronger conjecture on truncated Jacobi triple product series, arXiv:2411.13818 [math.NT], 2024. See p. 20.
G. H. Hardy, Divergent Series, Oxford University Press, 1949. p. 29. - _Johannes W. Meijer_, Oct 16 2009
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bull. Amer. Math. Soc., Vol. 50, No. 4 (2013), pp. 527-628, preprint, arXiv:1303.1856 [math.NT], 2013.
István Mezo, Gompertz constant, Gregory coefficients and a series of the logarithm function, Journal of Analysis & Number Theory, Vol. 2, No. 2 (2014), pp. 33-36.
Michael Penn, why some series are "regularizable", YouTube video (2023).
Simon Plouffe, -exp(1)*Ei(-1).
Tanguy Rivoal, On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant, Michigan Math. J., Vol. 61, No. 2 (2012), pp. 239-254.
Ed Sandifer, Divergent Series, How Euler Did It, MAA Online, June 2006. - _Johannes W. Meijer_, Oct 16 2009
Eric Weisstein's World of Mathematics, Gompertz Constant.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A000522 (arrangements), A001620, A000262, A002720, A002793, A058006 (alternating factorial sums), A091725, A099285, A153229, A201203, A245780, A283743 (Ei(1)/e), A321942, A369883.

SetDefaultRealField(RealField(100)); ExponentialIntegralE1(1)*Exp(1); // G. C. Greubel, Dec 04 2018

RealDigits[N[-Exp[1]*ExpIntegralEi[-1], 105]][[1]]
(* Second program: *)
G = 1/Fold[Function[2*#2 - #2^2/#1], 2, Reverse[Range[10^4]]] // N[#, 105]&; RealDigits[G] // First (* Jean-François Alcover, Sep 19 2014 *)

eint1(1)*exp(1) \\ Charles R Greathouse IV, Apr 23 2013

numerical_approx(exp_integral_e(1,1)*exp(1), digits=100) # G. C. Greubel, Dec 04 2018

phi(1) = e*(Sum_{k>=1} (-1)^(k-1)/(k*k!) - Gamma) = 0.596347362323194... where Gamma is the Euler constant.
G = 0.596347... = 1/(1+1/(1+1/(1+2/(1+2/(1+3/(1+3/(1+4/(1+4/(1+5/(1+5/(1+6/(... - Philippe Deléham, Aug 14 2005
Equals A001113*A099285. - Johannes W. Meijer, Oct 16 2009
From Peter Bala, Oct 11 2012: (Start)
Stieltjes found the continued fraction representation G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The numerators are in A002793 and the denominators in A002720.
Also, 1 - G has the continued fraction representation 1/(3 - 2/(5 - 6/(7 - ... -n*(n+1)/((2*n+3) - ...)))) with convergents beginning [1/3, 5/13, 29/73, 201/501, ...]. The numerators are in A201203 (unsigned) and the denominators are in A000262.
(End)
G = f(1) with f solution to the o.d.e. x^2*f'(x) + (x+1)*f(x)=1 such that f(0)=1. - Jean-François Alcover, May 28 2013
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..1} 1/(1-log(x)) dx.
Equals Integral_{x=1..oo} exp(1-x)/x dx.
Equals Integral_{x=0..oo} exp(-x)*log(x+1) dx.
Equals Integral_{x=0..oo} exp(-x)/(x+1) dx. (End)
From Gleb Koloskov, May 01 2021: (Start)
Equals Integral_{x=0..1} LambertW(e/x)-1 dx.
Equals Integral_{x=0..1} 1+1/LambertW(-1,-x/e) dx. (End)
Equals lim_{n->oo} A040027(n)/A000110(n+1). - Vaclav Kotesovec, Feb 22 2021
G = lim_{n->oo} A321942(n)/A000262(n). - Peter Bala, Mar 21 2022
Equals Sum_{n >= 1} 1/(n*L(n, -1)*L(n-1, -1)), where L(n, x) denotes the n-th Laguerre polynomial. This is the case x = 1 of the identity Integral_{t >= 0} exp(-t)/(x + t) dt = Sum_{n >= 1} 1/(n*L(n, -x)*L(n-1, -x)) valid for Re(x) > 0. - Peter Bala, Mar 21 2024
Equals lim_{n->oo} Sum_{k >= 0} (n/(n + 1))^k/(n + k). Cf. A099285. - Peter Bala, Jun 18 2024

Additional references from Gerald McGarvey, Oct 10 2005

Link corrected by Johannes W. Meijer, Aug 01 2009

A005990 a(n) = (n-1)*(n+1)!/6.

Original entry on oeis.org

0, 1, 8, 60, 480, 4200, 40320, 423360, 4838400, 59875200, 798336000, 11416204800, 174356582400, 2833294464000, 48819843072000, 889218570240000, 17072996548608000, 344661117825024000, 7298706024529920000, 161787983543746560000

Offset: 1

N. J. A. Sloane

Coefficients of Gandhi polynomials.
a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} max(pi(i)-i,0), i.e., the total positive displacement of all letters in all permutations on n letters. - Franklin T. Adams-Watters, Oct 25 2006
a(n) is also the sum of the excedances of all permutations of [n]. An excedance of a permutation p of [n] is an i (1 <= i <= n-1) such that p(i) > i. Proof: i is an excedance if p(i) = i+1, i+2, ..., n (n-i possibilities), with the remaining values of p forming any permutation of [n]\{p(i)} in the positions [n]\{i} ((n-1)! possibilities). Summation of i(n-i)(n-1)! over i from 1 to n-1 completes the proof. Example: a(3)=8 because the permutations 123, 132, 213, 231, 312, 321 have excedances NONE, {2}, {1}, {1,2}, {1}, {1}, respectively. - Emeric Deutsch, Oct 26 2008
a(n) is also the number of doubledescents in all permutations of {1,2,...,n-1}. We say that i is a doubledescent of a permutation p if p(i) > p(i+1) > p(i+2). Example: a(3)=8 because each of the permutations 1432, 4312, 4213, 2431, 3214, 3421 has one doubledescent, the permutation 4321 has two doubledescents and the remaining 17 permutations of {1,2,3,4} have no doubledescents. - Emeric Deutsch, Jul 26 2009
Equals the second right hand column of A167568 divided by 2. - Johannes W. Meijer, Nov 12 2009
Half of sum of abs(p(i+1) - p(i)) over all permutations on n, e.g., 42531 = 2 + 3 + 2 + 2 = 9, and the total over all permutations on {1,2,3,4,5} is 960. - Jon Perry, May 24 2013
a(n) gives the number of non-occupied corners in tree-like tableaux of size n+1 (see Gao et al. link). - Michel Marcus, Nov 18 2015
a(n) is the number of sequences of n+2 balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence. - Jeremy Dover, Sep 26 2017
a(n) is the number of triangles (3-cycles) in the (n+1)-alternating group graph. - Eric W. Weisstein, Jun 09 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..300
D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318. (Annotated scanned copy)
Alice L. L. Gao, Emily X. L. Gao, and Brian Y. Sun, Zubieta's Conjecture on the Enumeration of Corners in Tree-like Tableaux, arXiv:1511.05434 [math.CO], 2015. The second version of this paper has a different title and different authors: A. L. L. Gao, E. X. L. Gao, P. Laborde-Zubieta, and B. Y. Sun, Enumeration of Corners in Tree-like Tableaux and a Conjectural (a,b)-analogue, arXiv preprint arXiv:1511.05434v2, 2015.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Eric Weisstein's World of Mathematics, Alternating Group Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.

Cf. A001715, A090672, A167568.

Cf. A001113, A001620, A091725, A099285.

[(n-1)*Factorial(n+1)/6: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011

[ seq((n-1)*(n+1)!/6,n=1..40) ];
a:=n->sum(sum(sum(n!/6, j=1..n),k=-1..n),m=0..n): seq(a(n), n=0..19); # Zerinvary Lajos, May 11 2007
seq(sum(mul(j,j=3..n), k=3..n)/3, n=2..21); # Zerinvary Lajos, Jun 01 2007
restart: G(x):=x^3/(1-x)^2: f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/3!,n=2..21); # Zerinvary Lajos, Apr 01 2009

Table[Sum[n!/6, {i, 3, n}], {n, 2, 21}] (* Zerinvary Lajos, Jul 12 2009 *)
Table[(n - 1) (n + 1)!/6, {n, 20}] (* Harvey P. Dale, Apr 07 2019 *)
Table[(n - 1) Pochhammer[4, n - 2], {n, 20}] (* Eric W. Weisstein, Jun 09 2019 *)
Table[(n - 1) Gamma[n + 2]/6, {n, 20}] (* Eric W. Weisstein, Jun 09 2019 *)
Range[0, 20]! CoefficientList[Series[x/(1 - x)^4, {x, 0, 20}], x] (* Eric W. Weisstein, Jun 09 2019 *)

a(n)=(n-1)*(n+1)!/6 \\ Charles R Greathouse IV, May 24 2013

a(n) = A090672(n)/2.
a(n) = A052571(n+2)/6. - Zerinvary Lajos, May 11 2007
a(n) = Sum_{m=0..n} Sum_{k=-1..n} Sum_{j=1..n} n!/6, n >= 0. - Zerinvary Lajos, May 11 2007
If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j)) then a(n+1) = (-1)^(n-1)*f(n,1,-4), (n >= 1). - Milan Janjic, Mar 01 2009
E.g.f.: (-1+3*x)/(3!*(1-x)^3), a(0) = -1/3!. Such e.g.f. computations resulted from e-mail exchange with Gary Detlefs. - Wolfdieter Lang, May 27 2010
a(n) = ((n+3)!/2) * Sum_{j=i..k} (k+1)!/(k+3)!, with offset 0. - Gary Detlefs, Aug 05 2010
a(n) = (n+2)!*Sum_{k=1..n-1} 1/((2*k+4)*(k+3)). - Gary Detlefs, Oct 09 2011
a(n) = (n+2)!*(1 + 3*(H(n+1) - H(n+2)))/6, where H(n) is the n-th harmonic number. - Gary Detlefs, Oct 09 2011
With offset = 0, e.g.f.: x/(1-x)^4. - Geoffrey Critzer, Aug 30 2013
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=2} 1/a(n) = 3*(Ei(1) - gamma) - 6*e + 27/2, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=2} (-1)^n/a(n) = 3*(gamma - Ei(-1)) - 3/2, where Ei(-1) = -A099285. (End)

Better definition from Robert Newstedt

A062119 a(n) = n! * (n-1).

Original entry on oeis.org

0, 2, 12, 72, 480, 3600, 30240, 282240, 2903040, 32659200, 399168000, 5269017600, 74724249600, 1133317785600, 18307441152000, 313841848320000, 5690998849536000, 108840352997376000, 2189611807358976000, 46225138155356160000, 1021818843434188800000

Offset: 1

Olivier Gérard, Jun 13 2001

For n > 0, a(n) = number of permutations of length n+1 that have 2 predetermined elements nonadjacent; e.g., for n=2, the permutations with, say, 1 and 2 nonadjacent are 132 and 231, therefore a(2)=2. - Jon Perry, Jun 08 2003
Number of multiplications performed when computing the determinant of an n X n matrix by definition. - Mats Granvik, Sep 12 2008
Sum of the length of all cycles (excluding fixed points) in all permutations of [n]. - Olivier Gérard, Oct 23 2012
Number of permutations of n distinct objects (ABC...) 1 (one) times >>("-", A, AB, ABC, ABCD, ABCDE, ..., ABCDEFGHIJK, infinity) and one after the other to resemble motif: A (1) AB (1-1), AAB (2-1), AAAB (3-1), AAAAB (4-1), AAAAAB (5-1), AAAAAAB (6-1), AAAAAAAB (7-1), AAAAAAAAB (8-1) etc.,>> "1(one) fixed point". Example:motif: AAAB (or BBBA) 12 * one (1) fixed point etc. Let: AAAB ................ 'A'BCD 1. 'A'BDC 2. 'A'CBD 3. ACDB 'A'DBC 4. 'A'DCB B'A'CD 5. B'A'DC 6. BCAD 7. BCDA BD'A'C 8. BDCA C'A'BD 9. C'A'DB CB'A'D 10. CBDA CDAB CDBA D'A'BC 11. DACB DB'A'C 12. DBCA DCAB DCBA. - Zerinvary Lajos, Nov 27 2009 (does anybody understand what this is supposed to say? - Joerg Arndt, Jan 10 2015)
a(n) is the number of ways to arrange n books on two bookshelves so that each shelf receives at least one book. - Geoffrey Critzer, Feb 21 2010
a(n) = number whose factorial base representation (A007623) begins with digit {n-1} and is followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 10, 200, 3000, 40000, 500000, 6000000, 70000000, 800000000, 9000000000, A0000000000, B00000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015

Harry J. Smith, Table of n, a(n) for n = 1..100
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Ugbene Ifeanyichukwu Jeff, Ogundele Olaniyi Suraju, and Ndubuisi Rich Ugochukwu, Digraph of the full transformation semigroup, J. Disc. Math. Sci. Crypt. (2021).
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
S. R. Schmitt Determinants [From _Mats Granvik_, Sep 12 2008]
Index entries for sequences related to factorial base representation

Column 2 of A257503 (apart from initial zero. Equally, row 2 of A257505).
Cf. A001286 (same sequence divided by 2).
Cf. A000142, A007623, A018931, A052849.
Cf. A001563. - Zerinvary Lajos, Aug 27 2008
Cf. sequences with formula (n + k)*n! listed in A282466.
Cf. A001113, A001620, A068985, A091725, A099285.

a062119 n = (n - 1) * a000142 n  -- Reinhard Zumkeller, Aug 27 2012

G(x):=x^2/(1-x)^2: f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..19); # Zerinvary Lajos, Apr 01 2009

a[n_]:=1*(n+2)!-2*(n+1)!; (* or *) a[n_]:=n!*(n-1); (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)
Table[n!(n-1),{n,20}] (* Harvey P. Dale, Aug 29 2021 *)

{ f=1; for (n=1, 100, f*=n; write("b062119.txt", n, " ", f*(n - 1)) ) } \\ Harry J. Smith, Aug 02 2009

a(n) = n!*(n-1); \\ Altug Alkan, May 04 2018

(define (A062119 n) (* (- n 1) (A000142 n))) ;; Antti Karttunen, May 07 2015

a(n) = n! * (n-1).
E.g.f.: x^2/(1-x)^2. - Geoffrey Critzer, Feb 21 2010
a(n) = 2 * A001286(n).
a(n) = A001563(n) - A000142(n). - Antti Karttunen, May 07 2015, hinted by crossref left by Lajos.
From Amiram Eldar, Jul 11 2020: (Start)
Sum_{n>=2} 1/a(n) = Ei(1) + 2 - e - gamma = A091725 + 2 - A001113 - A001620.
Sum_{n>=2} (-1)^n/a(n) = gamma - Ei(-1) - 1/e = A001620 + A099285 - A068985. (End)

Last term a(19) corrected by Harry J. Smith, Aug 02 2009

A001809 a(n) = n! * n(n-1)/4.

Original entry on oeis.org

0, 0, 1, 9, 72, 600, 5400, 52920, 564480, 6531840, 81648000, 1097712000, 15807052800, 242853811200, 3966612249600, 68652904320000, 1255367393280000, 24186745110528000, 489781588488192000, 10400656084955136000, 231125690776780800000, 5364548928029491200000

Offset: 0

N. J. A. Sloane

a(n) = n!*n*(n-1)/4 gives the total number of inversions in all the permutations of [n]. [Stern, Terquem] Proof: For fixed i,j and for fixed I,J (i < j, I > J, 1 <= i,j,I,J <= n), we have (n-2)! permutations p of [n] for which p(i)=I and p(j)=J (permute {1,2,...,n} \ {I,J} in the positions (1,2,...,n) \ {i,j}). There are n*(n-1)/2 choices for the pair (i,j) with i < j and n*(n-1)/2 choices for the pair (I,J) with I > J. Consequently, the total number of inversions in all the permutations of [n] is (n-2)!*(n*(n-1)/2)^2 = n!*n*(n-1)/4. - Emeric Deutsch, Oct 05 2006
To state this another way, a(n) is the number of occurrences of the pattern 12 in all permutations of [n]. - N. J. A. Sloane, Apr 12 2014
Equivalently, this is the total Denert index of all permutations on n letters (cf. A008302). - N. J. A. Sloane, Jan 20 2014
Also coefficients of Laguerre polynomials. a(n)=A021009(n,2), n >= 2.
a(n) is the number of edges in the Cayley graph of the symmetric group S_n with respect to the generating set consisting of transpositions. - Avi Peretz (njk(AT)netvision.net.il), Feb 20 2001
a(n+1) is the sum of the moments over all permutations of n. E.g. a(4) is [1,2,3].[1,2,3] + [1,3,2].[1,2,3] + [2,1,3].[1,2,3] + [2,3,1].[1,2,3] + [3,1,2].[1,2,3] + [3,2,1].[1,2.3] = 14 + 13 + 13 + 11 + 11 + 10 = 72. - Jon Perry, Feb 20 2004
Derivative of the q-factorial [n]!, evaluated at q=1. Example: a(3)=9 because (d/dq)[3]!=(d/dq)((1+q)(1+q+q^2))=2+4q+3q^2 is equal to 9 at q=1. - Emeric Deutsch, Apr 19 2007
Also the number of maximal cliques in the n-transposition graph for n > 1. - Eric W. Weisstein, Dec 01 2017
a(n-1) is the number of trees on [n], rooted at 1, with exactly two leaves.  A leaf is a non-root vertex of degree 1. - Nikos Apostolakis, Dec 27 2021

			G.f. = x^2 + 9*x^3 + 72*x^4 + 600*x^5 + 5400*x^6 + 52920*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
Simon Altmann and Eduardo L. Ortiz, Editors, Mathematical and Social Utopias in France: Olinde Rodrigues and His Times, Amer. Math. Soc., 2005.
David M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 90.
Cornelius Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
Edward M. Reingold, Jurg Nievergelt and Narsingh Deo, Combinatorial Algorithms, Prentice-Hall, 1977, section 7.1, p. 287.
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).
Olry Terquem, Liouville's Journal, 1838.

T. D. Noe, Table of n, a(n) for n=0..100
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].
Eric Babson and Einar Steingrimsson, Generalized Permutation Patterns and Classification of the Mahonian Statistics, Séminaire Lotharingien de Combinatoire, B44b (2000), 18 pp.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
FindStat - Combinatorial Statistic Finder, The Denert index of a permutation
Dominique Foata and Marcel-Paul Schützenberger, Major Index and inversion number of permutations , Math. Nachr. 83 (1978), 143-159
Dexter Jane L. Indong and Gilbert R. Peralta, Inversions of permutations in Symmetric, Alternating, and Dihedral Groups, JIS, Vol. 11 (2008), Article 08.4.3.
Warren P. Johnson, Review of Altmann-Ortiz book, Amer. Math. Monthly, Vol. 114, No. 8 (2007), pp. 752-758.
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
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)
M. Stern, Aufgaben, Journal für die reine und angewandte Mathematik, Vol. 18 (1838), p. 100.
Thotsaporn Thanatipanonda, Inversions and major index for permutations, Math. Mag., Vol. 77, No. 2 (April 2004), pp. 136-140.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Transposition Graph.
Index entries for sequences related to Laguerre polynomials.

Cf. A034968 (the inversion numbers of permutations listed in alphabetic order). See also A053495 and A064038.

Cf. A001620, A008302, A099285.

[Factorial(n)*n*(n-1)/4: n in [0..20]]; // Vincenzo Librandi, Jun 15 2015

A001809 := n->n!*n*(n-1)/4;
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=1)}, labeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=0..19); # Zerinvary Lajos, Feb 07 2008
with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*binomial(n,2)/2, n=0..21); # Zerinvary Lajos, Jun 11 2008

Table[n! n (n - 1)/4, {n, 0, 18}]
Table[n! Binomial[n, 2]/2, {n, 0, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
Coefficient[Table[n! LaguerreL[n, x], {n, 20}], x, 2] (* Eric W. Weisstein, Dec 01 2017 *)

PARI
```
{a(n) = n! * n * (n-1) / 4};
```

[factorial(m) * binomial(m, 2) / 2 for m in range(19)]  # Zerinvary Lajos, Jul 05 2008

E.g.f.: (1/2)*x^2/(1-x)^3.
a(n) = a(n-1)*n^2/(n-2), n > 2; a(2)=1.
a(n) = n*a(n-1) + (n-1)!*n*(n-1)/2, a(1) = 0, a(2) = 1; a(n) = sum (first n! terms of A034968); a(n) = sum of the rises j of permutations (p(j)If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k}(C(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j))) then a(n)=(-1)^n*f(n,2,-3), (n>=2). - Milan Janjic, Mar 01 2009
a(n) = Sum_k k*A008302(n,k). - N. J. A. Sloane, Jan 20 2014
a(n+2) = n*n!*(n+1)^2 / 4 = A000142(n) * (A000292(n) + A000330(n))/2 = sum of the cumulative sums of all the permutations of numbers from 1 to n, where A000142(n) = n! and sequences A000292(n) and A000330(n) are sequences of minimal and maximal values of cumulative sums of all the permutations of numbers from 1 to n. - Jaroslav Krizek, Sep 13 2014
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=2} 1/a(n) = 12 - 4*e.
Sum_{n>=2} (-1)^n/a(n) = 8*gamma - 4 - 4/e - 8*Ei(-1), where gamma is Euler's constant (A001620) and -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

More terms and new description from Michael Somos, May 19 2000

Simpler description from Emeric Deutsch, Oct 05 2006

A001754 Lah numbers: a(n) = n!*binomial(n-1,2)/6.

Original entry on oeis.org

0, 0, 1, 12, 120, 1200, 12600, 141120, 1693440, 21772800, 299376000, 4390848000, 68497228800, 1133317785600, 19833061248000, 366148823040000, 7113748561920000, 145120470663168000, 3101950060425216000, 69337707233034240000, 1617879835437465600000

Offset: 1

N. J. A. Sloane

a(n+1) = Sum_{pi in Symm(n)} Sum_{i=1..n} max(pi(i)-i,0)^2, i.e., the sum of the squares of the positive displacement of all letters in all permutations on n letters. - Franklin T. Adams-Watters, Oct 25 2006

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
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).

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Column 3 of A008297.
Column m=3 of unsigned triangle A111596.
Cf. A005990, A053495, A260665.
Cf. A001113, A001620, A091725, A099285.

[Factorial(n)*Binomial(n-1, 2)/6: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011

Maple
```
[seq(n!*binomial(n-1,2)/6, n=1..40)];
```

Table[(n-2)*(n-1)*n!/12, {n, 21}] (* Arkadiusz Wesolowski, Nov 26 2012 *)
With[{nn=30},CoefficientList[Series[(x/(1-x))^3/6,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 04 2017 *)

[factorial(n-1)*binomial(n,3)/2 for n in (1..30)] # G. C. Greubel, May 10 2021

E.g.f.: ((x/(1-x))^3)/3!.
If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i)*x^(k-j)) then a(n+1) = (-1)^n*f(n,2,-4), n >= 2. - Milan Janjic, Mar 01 2009
a(n) = Sum_{k>=1} k * A260665(n,k). - Alois P. Heinz, Nov 14 2015
D-finite with recurrence (-n+5)*a(n) + (n-2)*(n-3)*a(n-1) = 0, n >= 4. - R. J. Mathar, Jan 06 2021
From Amiram Eldar, May 02 2022: (Start)
Sum_{n>=3} 1/a(n) = 6*(gamma - Ei(1)) + 9, where gamma = A001620 and Ei(1) = A091725.
Sum_{n>=3} (-1)^(n+1)/a(n) = 18*(gamma - Ei(-1)) - 12/e - 9, where Ei(-1) = -A099285 and e = A001113. (End)

A074143 a(1) = 1; a(n) = n * Sum_{k=1..n-1} a(k).

Original entry on oeis.org

1, 2, 9, 48, 300, 2160, 17640, 161280, 1632960, 18144000, 219542400, 2874009600, 40475635200, 610248038400, 9807557760000, 167382319104000, 3023343138816000, 57621363351552000, 1155628453883904000, 24329020081766400000, 536454892802949120000

Offset: 1

Amarnath Murthy, Aug 28 2002

nonn

a(n) is also the number of elements of the alternating semigroup (A^c_n) for F(n, p) if p = n - 1 (cf. A001710). - Bakare Gatta Naimat, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets
Stephen Lipscomb, Symmetric inverse semigroups, Mathematical surveys and monographs, Vol. 46 Amer. Math. Soc. (1996).
Michael Penn, Australian Mathematical Olympiad 2018 Question 5, Youtube video, 2020.

A diagonal of A254040.
Cf. A001563, A001710.
Cf. A007808, A082425, A082427, A082428, A082430, A383436, A383437.
Cf. A001620, A091725, A099285.

[Numerator(Factorial(n)/2*n): n in [1..30]]; // Vincenzo Librandi, Apr 15 2014

seq(sum(mul(j,j=3..n), k=1..n), n=1..19); # Zerinvary Lajos, Jun 01 2007
a := n -> `if`(n=1,1,n!*n/2): seq(a(n), n=1..19); # Peter Luschny, Jan 22 2016

A074143[1] = 1; A074143[n_] := A074143[n] = n * Sum[a[k], {k, n - 1}]; Array[A074143, 20] (* T. D. Noe, Apr 05 2011 *)
Table[Numerator[n!/2 n], {n, 21}] (* Vincenzo Librandi, Apr 15 2014 *)

def b(n): return 1/2 if (n==1) else n^2*b(n-1)/(n-1)
def A074143(n): return b(n) + int(n==1)/2
[A074143(n) for n in range(1,41)] # G. C. Greubel, Nov 29 2022

a(n) = n^2 * a(n-1)/(n-1) for n > 2.
a(n) = n*ceiling(n!/2) = n*A001710(n) = ceiling(A001563(n)/2). - Henry Bottomley, Nov 27 2002
a(n) = ((n+1)!-n!)/2 for n > 1. - Vladimir Joseph Stephan Orlovsky, Apr 03 2011
G.f.: (U(0) + x)/(2*x) where U(k) = 1 - 1/(k+1 - x*(k+1)^2*(k+2)/(x*(k+1)*(k+2) - 1/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 27 2012
G.f.: 1/2 + Q(0), where Q(k)= 1 - 1/(k+2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3)-1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
a(n) = Sum_{j = 0..n} (-1)^(n-j)*binomial(n, j)*(j)^(n+1) / (n+1), n > 1, a(1) = 1. - Vladimir Kruchinin, Jun 01 2013
a(n) = numerator(n!/2*n). - Vincenzo Librandi, Apr 15 2014
a(n) is F(n;p) = n^2(n-1)!/2 if p = n-1 in A^c_n. For instance for n=4 and p=n-1: F(4; 4-1)= 4^2(4-1)!/2 = 16*6/2 = 48. - Bakare Gatta Naimat, Nov 18 2015
From Seiichi Manyama, Apr 27 2025: (Start)
E.g.f.: x/2 * (1 + 1/(1-x)^2).
a(n) = (n+2) * a(n-1) - (n-1) * a(n-2) for n > 3. (End)
From Amiram Eldar, May 04 2025: (Start)
Sum_{n>=1} 1/a(n) = 2*ExpIntegralEi(1) - 2*gamma - 1 = 2*A091725 - 2*A001620 - 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*gamma - 1 - 2*ExpIntegralEi(-1) = 2*A001620 - 1 + 2*A099285. (End)

More terms from Henry Bottomley, Nov 27 2002

A005461 Number of simplices in barycentric subdivision of n-simplex.

Original entry on oeis.org

1, 15, 180, 2100, 25200, 317520, 4233600, 59875200, 898128000, 14270256000, 239740300800, 4249941696000, 79332244992000, 1556132497920000, 32011868528640000, 689322235650048000, 15509750302126080000, 364022962973429760000, 8898339094906060800000

Offset: 1

N. J. A. Sloane

			G.f. = x + 15*x^2 + 180*x^3 + 2100*x^4 + 25200*x^5 + 317520*x^6 + ...

R. Austin, R. K. Guy, and R. Nowakowski, unpublished notes, circa 1987.
R. K. Guy, personal communication.
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 = 1..440
R. Austin, R. K. Guy, and R. Nowakowski, Unpublished notes, 1987.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics, Vol. 10, No. 7 (2022), 1161.

Cf. A001113, A001620, A028246, A091725, A099285.
Cf. A005460, A005462, A005463, A005464, A005465.
Cf. A001297.

[Factorial(n-1)*StirlingSecond(n+3,n): n in [1..35]]; // G. C. Greubel, Nov 23 2022

a:=n->sum((n-j)*n!/4!, j=3..n): seq(a(n), n=4..17); # Zerinvary Lajos, Apr 29 2007

Table[(n(n+1)(n+3)!)/48,{n,20}] (* Harvey P. Dale, Mar 14 2012 *)
a[ n_] := If[ n < 0, 0, n (n + 1) (n + 3)! / 48]; (* Michael Somos, May 27 2014 *)

[factorial(m+1)*binomial(m-1,2)/24 for m in range(3, 19)] # Zerinvary Lajos, Jul 05 2008

[binomial(n,4)*factorial (n-2)/2 for n in range(4, 18)] #  Zerinvary Lajos, Jul 07 2009

a(n) = n*(n + 1)*(n + 3)!/48.
Essentially Stirling numbers of second kind - see A028246.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n-3) = (-1)^n*f(n,4,-3), (n>=4). - Milan Janjic, Mar 01 2009
E.g.f.: t*(3*t + 2)/(2*(t - 1)^6). - Ran Pan, Jul 10 2016
a(n) ~ sqrt(Pi/2)*exp(-n)*n^(n+1/2)*(n^5/24 + 85*n^4/288 + 5065*n^3/6912 + 955841*n^2/1244160 + 3710929*n/11943936). - Ilya Gutkovskiy, Jul 10 2016
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 16*(e + gamma - Ei(1)) - 64/3, where e = A001113, gamma = A001620, and Ei(1) = A091725.
Sum_{n>=1} (-1)^(n+1)/a(n) = 32*(gamma - Ei(-1)) - 16/e - 56/3, where Ei(-1) = -A099285. (End)
a(n) = (n-1)! * Stirling2(n+3, n). - G. C. Greubel, Nov 23 2022

More terms from Harvey P. Dale, Mar 14 2012

A061206 a(n) = total number of occurrences of the consecutive pattern 1324 in all permutations of [n+3].

Original entry on oeis.org

1, 10, 90, 840, 8400, 90720, 1058400, 13305600, 179625600, 2594592000, 39956716800, 653837184000, 11333177856000, 207484333056000, 4001483566080000, 81096733605888000, 1723305589125120000, 38318206628782080000, 889833909490606080000, 21543347282404147200000

Offset: 1

Melvin J. Knight (knightmj(AT)juno.com), May 30 2001

nonn

a(n) is the number of sequences of n+3 balls colored with at most n colors such that exactly four balls are the same color as some other ball in the sequence. - Jeremy Dover, Sep 27 2017

			a(4)=840 because 4*(7!)/24 = 4*7*6*5 = 840.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Milan Janjić, Enumerative Formulas for Some Functions on Finite Sets.

Cf. A000142, A001286, A001339, A001563, A005990, A264173.

Cf. A001620, A091725, A099285.

[n*Factorial(n+3)/24: n in [1..20]]; // Vincenzo Librandi, Oct 11 2011

a := n -> n!*binomial(-n,4): seq(a(n),n=1..20); # Peter Luschny, Apr 29 2016

Array[# (# + 3)!/24 &, 20] (* or *) Array[#!*Binomial[-#, 4] &, 20] (* Michael De Vlieger, Sep 30 2017 *)

a(n) = n*(n+3)!/24; \\ Altug Alkan, Oct 08 2017

[binomial(n,4)*factorial (n-3) for n in range(4, 21)] # Zerinvary Lajos, Jul 07 2009

a(n) = n*(n+3)!/24.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i) * x^(k-j), then a(n-3) = (-1)^n*f(n,4,-2), (n >= 4). - Milan Janjic, Mar 01 2009
E.g.f.: x/(1-x)^5. (This was initiated by e-mail exchange with Gary Detlefs.) - Wolfdieter Lang, May 28 2010
a(n) = ((n+4)!/6) * Sum_{k=1..n} (k+2)!/(k+4)!. - Gary Detlefs, Aug 05 2010
a(n) = Sum_{k>0} k * A264173(n+3,k). - Alois P. Heinz, Nov 06 2015
a(n) = n!*binomial(-n,4). - Peter Luschny, Apr 29 2016
From Amiram Eldar, Sep 24 2022: (Start)
Sum_{n>=1} 1/a(n) = 118/3 - 16*e - 4*gamma + 4*Ei(1), where gamma is Euler's constant (A001620) and Ei(1) is the exponential integral at 1 (A091725).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2/3 - 8/e + 4*gamma - 4*Ei(-1), where -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

More terms from Jason Earls, Jun 12 2001
Corrected by Zerinvary Lajos, Jul 07 2009
More precise definition from Alois P. Heinz, Nov 06 2015

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cp's OEIS Frontend

A163931 Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1.

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A001286 Lah numbers: a(n) = (n-1)*n!/2.

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A073003 Decimal expansion of -exp(1)*Ei(-1), also called Gompertz's constant, or the Euler-Gompertz constant.

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A005990 a(n) = (n-1)*(n+1)!/6.

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A062119 a(n) = n! * (n-1).

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