A087981 -id:A087981 - cp's OEIS frontend

A010027 Triangle read by rows: T(n,k) is the number of permutations of [n] having k consecutive ascending pairs (0 <= k <= n-1).

1, 1, 1, 1, 2, 3, 1, 3, 9, 11, 1, 4, 18, 44, 53, 1, 5, 30, 110, 265, 309, 1, 6, 45, 220, 795, 1854, 2119, 1, 7, 63, 385, 1855, 6489, 14833, 16687, 1, 8, 84, 616, 3710, 17304, 59332, 133496, 148329, 1, 9, 108, 924, 6678, 38934, 177996, 600732, 1334961, 1468457, 1

Offset: 1

N. J. A. Sloane

A "consecutive ascending pair" in a permutation p_1, p_2, ..., p_n is a pair p_i, p_{i+1} = p_i + 1.
From Emeric Deutsch, May 15 2010: (Start)
The same triangle, but with rows indexed differently, also arises as follows: U(n,k) = number of permutations of [n] having k blocks (1 <= k <= n), where a block of a permutation is a maximal sequence of consecutive integers which appear in consecutive positions. For example, the permutation 5412367 has 4 blocks: 5, 4, 123, and 67.
When seen as coefficients of polynomials with decreasing exponents: evaluations are A001339 (x=2), A081923 (x=3), A081924 (x=4), A087981 (x=-1).
The sum of the entries in row n is n!.
U(n,n) = A000255(n-1) = d(n-1) + d(n), U(n,n-1)=d(n), where d(j)=A000166(j) (derangement numbers). (End)
This is essentially the reversal of the exponential Riordan array [exp(-x)/(1-x)^2,x] (cf. A123513). - Paul Barry, Jun 17 2010
U(n-1, k-2) * n*(n-1)/k = number of permutations of [n] with k elements not fixed by the permutation. - Michael Somos, Aug 19 2018

			Triangle starts:
  1;
  1, 1;
  1, 2,   3;
  1, 3,   9,  11;
  1, 4,  18,  44,   53;
  1, 5,  30, 110,  265,   309;
  1, 6,  45, 220,  795,  1854,   2119;
  1, 7,  63, 385, 1855,  6489,  14833,  16687;
  1, 8,  84, 616, 3710, 17304,  59332, 133496,  148329;
  1, 9, 108, 924, 6678, 38934, 177996, 600732, 1334961, 1468457;
  ...
For n=3, the permutations 123, 132, 213, 231, 312, 321 have respectively 2,0,0,1,1,0 consecutive ascending pairs, so row 3 of the triangle is 3,2,1. - _N. J. A. Sloane_, Apr 12 2014
In the alternative definition, T(4,2)=3 because we have 234.1, 4.123, and 34.12 (the blocks are separated by dots). - _Emeric Deutsch_, May 16 2010

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

Alois P. Heinz, Rows n = 1..150, flattened
A. N. Myers, Counting permutations by their rigid patterns, J. Combin. Theory, A 99 (2002), 345-357. [_Emeric Deutsch_, May 15 2010]

Diagonals, reading from the right-hand edge: A000255, A000166, A000274, A000313, A001260, A001261. A045943 is another diagonal.
Cf. A123513 (mirror image).
A289632 is the analogous triangle with the additional restriction that all consecutive pairs must be isolated, i.e., must not be back-to-back to form longer consecutive sequences.

U := proc (n, k) options operator, arrow: factorial(k+1)*binomial(n, k)*(sum((-1)^i/factorial(i), i = 0 .. k+1))/n end proc: for n to 10 do seq(U(n, k), k = 1 .. n) end do; # yields sequence in triangular form; # Emeric Deutsch, May 15 2010

t[n_, k_] := Binomial[n, k]*Subfactorial[k+1]/n; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 07 2014, after Emeric Deutsch *)
T[0,0]:=0; T[1,1]:=1; T[n_,n_]:=T[n,n]=(n-1)T[n-1,n-1]+(n-2)T[n-2,n-2]; T[n_,k_]:=T[n,k]=T[n-1,k] (n-1)/(n-k); Flatten@Table[T[n,k],{n,1,10},{k,1,n}] (* Oliver Seipel, Dec 01 2024 *)

E.g.f.: exp(x*(y-1))/(1-x)^2. - Vladeta Jovovic, Jan 03 2003
From Emeric Deutsch, May 15 2010: (Start)
U(n,k) = binomial(n-1,k-1)*(k-1)!*Sum_{j=0..k-1} (-1)^(k-j-1)*(j+1)/(k-j-1)! (1 <= k <= n).
U(n,k) = (k+1)!*binomial(n,k)*(1/n)*Sum_{i=0..k+1} (-1)^i/i!.
U(n,k) = (1/n)*binomial(n,k)*d(k+1), where d(j)=A000166(j) (derangement numbers). (End)

More terms from Vladeta Jovovic, Jan 03 2003

Original definition from David, Kendall and Barton restored by N. J. A. Sloane, Apr 12 2014

A000023 Expansion of e.g.f. exp(-2*x)/(1-x).

Original entry on oeis.org

1, -1, 2, -2, 8, 8, 112, 656, 5504, 49024, 491264, 5401856, 64826368, 842734592, 11798300672, 176974477312, 2831591702528, 48137058811904, 866467058876416, 16462874118127616, 329257482363600896, 6914407129633521664, 152116956851941670912

Offset: 0

N. J. A. Sloane

A010843, A000023, A000166, A000142, A000522, A010842, A053486, A053487 are successive binomial transforms with the e.g.f. exp(k*x)/(1-x) and recurrence b(n) = n*b(n-1)+k^n and are related to incomplete gamma functions at k. In this case k=-2, a(n) = n*a(n-1)+(-2)^n = Gamma(n+1,k)*exp(k) = Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*i^(n-i)*(i+k)^i. - Vladeta Jovovic, Aug 19 2002

a(n) is the permanent of the n X n matrix with -1's on the diagonal and 1's elsewhere. - Philippe Deléham, Dec 15 2003

			G.f. = 1 - x + 2*x^2 - 2*x^3 + 8*x^4 + 8*x^5 + 112*x^6 + 656*x^7 + ... - _Michael Somos_, Nov 20 2018

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
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).

Simon Plouffe, Table of n, a(n) for n = 0..250
A. R. Kräuter, Permanenten - Ein kurzer Überblick, Séminaire Lotharingien de Combinatoire, B09b (1983), 34 pp.
A. R. Kräuter, Über die Permanente gewisser zirkulanter Matrizen und damit zusammenhängender Toeplitz-Matrizen, Séminaire Lotharingien de Combinatoire, B11b (1984), 11 pp.

Cf. A087891, A008290, A089258.

Cf. also A010843, A000166, A000142, A000522, A010842, A053486, A053487.

a000023 n = foldl g 1 [1..n]
  where g n m = n*m + (-2)^m
-- James Spahlinger, Oct 08 2012

a := n -> n!*add(((-2)^k/k!), k=0..n): seq(a(n), n=0..27); # Zerinvary Lajos, Jun 22 2007
seq(simplify(KummerU(-n, -n, -2)), n = 0..22); # Peter Luschny, May 10 2022

FoldList[#1*#2 + (-2)^#2 &, 1, Range[22]] (* Robert G. Wilson v, Jul 07 2012 *)
With[{r = Round[n!/E^2 - (-2)^(n + 1)/(n + 1)]}, r - (-1)^n Mod[(-1)^n r, 2^(n + Ceiling[-(2/n) - Log[2, n]])]] (* Stan Wagon May 02 2016 *)
a[n_] := (-1)^n x D[1/x Exp[x], {x, n}] x^n Exp[-x]
Table[a[n] /. x -> 2, {n, 0, 22}](* Gerry Martens , May 05 2016 *)

a(n)=if(n<0,0,n!*polcoeff(exp(-2*x+x*O(x^n))/(1-x),n))

my(x='x+O('x^66)); Vec( serlaplace( exp(-2*x)/(1-x)) ) \\ Joerg Arndt, Oct 06 2013

from sympy import exp, uppergamma
def A000023(n):
    return exp(-2) * uppergamma(n + 1, -2)  # David Radcliffe, Aug 20 2025

@CachedFunction
def A000023(n):
    if n == 0: return 1
    return n * A000023(n-1) + (-2)**n
[A000023(i) for i in range(23)]   # Peter Luschny, Oct 17 2012

a(n) = Sum_{k=0..n} A008290(n,k)*(-1)^k. - Philippe Deléham, Dec 15 2003
a(n) = Sum_{k=0..n} (-2)^(n-k)*n!/(n-k)! = Sum_{k=0..n} binomial(n, k)*k!*(-2)^(n-k). - Paul Barry, Aug 26 2004
a(n) = exp(-2)*Gamma(n+1,-2)  (incomplete Gamma function). - Mark van Hoeij, Nov 11 2009
a(n) = b such that (-1)^n*Integral_{x=0..2} x^n*exp(x) dx = c + b*exp(2). - Francesco Daddi, Aug 01 2011
G.f.: hypergeom([1,k],[],x/(1+2*x))/(1+2*x) with k=1,2,3 is the generating function for A000023, A087981, and A052124. - Mark van Hoeij, Nov 08 2011
D-finite with recurrence: - a(n) + (n-2)*a(n-1) + 2*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
E.g.f.: 1/E(0) where E(k) = 1 - x/(1-2/(2-(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+1)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: 1/Q(0), where Q(k) = 1 - x*(2*k-1) - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(n, k)*!k, where !k is the subfactorial A000166. a(n) = (-2)^n*hypergeom([1, -n], [], 1/2). - Vladimir Reshetnikov, Oct 18 2015
For n >= 3, a(n) = r - (-1)^n mod((-1)^n r, 2^(n - floor((2/n) + log_2(n)))) where r = {n! * e^(-2) - (-2)^(n+1)/(n+1)}. - Stan Wagon, May 02 2016
0 = +a(n)*(+4*a(n+1) -2*a(n+3)) + a(n+1)*(+4*a(n+1) +3*a(n+2) -a(n+3)) +a(n+2)*(+a(n+2)) if n>=0. - Michael Somos, Nov 20 2018
a(n) = KummerU(-n, -n, -2). - Peter Luschny, May 10 2022

A132159 Lower triangular matrix T(n,j) for double application of an iterated mixed order Laguerre transform inverse to A132014. Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-2-n,x).

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 24, 18, 6, 1, 120, 96, 36, 8, 1, 720, 600, 240, 60, 10, 1, 5040, 4320, 1800, 480, 90, 12, 1, 40320, 35280, 15120, 4200, 840, 126, 14, 1, 362880, 322560, 141120, 40320, 8400, 1344, 168, 16, 1, 3628800, 3265920, 1451520, 423360, 90720, 15120

Offset: 0

Tom Copeland, Nov 01 2007

The matrix operation b = T*a can be characterized several ways in terms of the coefficients a(n) and b(n), their o.g.f.'s A(x) and B(x), or their e.g.f.'s EA(x) and EB(x).
1) b(n) = n! Lag[n,(.)!*Lag[.,a1(.),-1],0], umbrally,
where a1(n) = n! Lag[n,(.)!*Lag[.,a(.),-1],0]
2) b(n) = (-1)^n * n! * Lag(n,a(.),-2-n)
3) b(n) = Sum_{j=0..n} (-1)^j * binomial(n,j) * binomial(-2,j) * j! * a(n-j)
4) b(n) = Sum_{j=0..n} binomial(n,j) * (j+1)! * a(n-j)
5) B(x) = (1-xDx))^(-2) A(x), formally
6) B(x) = Sum_{j>=0} (-1)^j * binomial(-2,j) * (xDx)^j A(x)
= Sum_{j>=0} (j+1) * (xDx)^j A(x)
7) B(x) = Sum_{j>=0} (j+1) * x^j * D^j * x^j A(x)
8) B(x) = Sum_{j>=0} (j+1)! * x^j * Lag(j,-:xD:,0) A(x)
9) EB(x) = Sum_{j>=0} x^j * Lag[j,(.)! * Lag[.,a1(.),-1],0]
10) EB(x) = Sum_{j>=0} Lag[j,a1(.),-1] * (-x)^j / (1-x)^(j+1)
11) EB(x) = Sum_{j>=0} x^n * Sum_{j=0..n} (j+1)!/j! * a(n-j) / (n-j)!
12) EB(x) = Sum_{j>=0} (-x)^j * Lag[j,a(.),-2-j]
13) EB(x) = exp(a(.)*x) / (1-x)^2 = (1-x)^(-2) * EA(x)
14) T = A094587^2 = A132013^(-2) = A132014^(-1)
where Lag(n,x,m) are the Laguerre polynomials of order m, D the derivative w.r.t. x and (:xD:)^j = x^j * D^j. Truncating the D operator series at the j = n term gives an o.g.f. for b(0) through b(n).
c = (1!,2!,3!,4!,...) is the sequence associated to T under the list partition transform and associated operations described in A133314. Thus T(n,k) = binomial(n,k)*c(n-k) . c are also the coefficients in formulas 4 and 8.
The reciprocal sequence to c is d = (1,-2,2,0,0,0,...), so the inverse of T is TI(n,k) = binomial(n,k)*d(n-k) = A132014. (A121757 is the reverse of T.)
These formulas are easily generalized for m applications of the basic operator n! Lag[n,(.)!*Lag[.,a(.),-1],0] by replacing 2 by m in formulas 2, 3, 5, 6, 12, 13 and 14, or (j+1)! by (m-1+j)!/(m-1)! in 4, 8 and 11. For further discussion of repeated applications of T, see A132014.
The row sums of T = [formula 4 with a(n) all 1] = [binomial transform of c] = [coefficients of B(x) with A(x) = 1/(1-x)] = A001339. Therefore the e.g.f. of A001339 = [formula 13 with a(n) all 1] = exp(x)*(1-x)^(-2) = exp(x)*exp[c(.)*x)] = exp[(1+c(.))*x].
Note the reciprocal is 1/{exp[(1+c(.))*x]} = exp(-x)*(1-x)^2 = e.g.f. of signed A002061 with leading 1 removed], which makes A001339 and the signed, shifted A002061 reciprocal arrays under the list partition transform of A133314.
The e.g.f. for the row polynomials (see A132382) implies they form an Appell sequence (see Wikipedia). - Tom Copeland, Dec 03 2013
As noted in item 12 above and reiterated in the Bala formula below, the e.g.f. is e^(x*t)/(1-x)^2, and the Poisson-Charlier polynomials P_n(t,y) have the e.g.f. (1+x)^y e^(-xt) (Feinsilver, p. 5), so the row polynomials R_n(t) of this entry are (-1)^n P_n(t,-2). The associated Appell sequence IR_n(t) that is the umbral compositional inverse of this entry's polynomials has the e.g.f. (1-x)^2 e^(xt), i.e., the e.g.f. of A132014 (noted above), and, therefore, the row polynomials (-1)^n PC(t,2). As umbral compositional inverses, R_n(IR.(t)) = t^n =  IR_n(R.(t)), where, by definition, P.(t)^n = P_n(t), is the umbral evaluation. - Tom Copeland, Jan 15 2016
T(n,k) is the number of ways to place (n-k) rooks in a 2 x (n-1) Ferrers board (or diagram) under the Goldman-Haglund i-row creation rook mode for i=2. Triangular recurrence relation is given by T(n,k) = T(n-1,k-1) + (n+1-k)*T(n-1,k). - Ken Joffaniel M. Gonzales, Jan 21 2016

			First few rows of the triangle are
    1;
    2,  1;
    6,  4,  1;
   24, 18,  6, 1;
  120, 96, 36, 8, 1;

Nathaniel Johnston, Rows 0..100, flattened
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
P. Feinsilver, Lie algebras, representations, and analytic semigroups through dual vector fields
Jay Goldman and James Haglund, Generalized rook polynomials, J. Combin. Theory A 91 (2000), 509-530.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Wikipedia, Appell sequence
Wikipedia, Sheffer sequence

Cf. A008277, A094587, A132013, A132382.

Columns: A000142 (k=0), A001563 (k=1), A001286 (k=2), A005990 (k=3), A061206 (k=4), A062199 (k=5), A062148 (k=6).

a132159 n k = a132159_tabl !! n !! k
a132159_row n = a132159_tabl !! n
a132159_tabl = map reverse a121757_tabl
-- Reinhard Zumkeller, Mar 06 2014

/* As triangle */ [[Binomial(n,k)*Factorial(n-k+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Feb 10 2016

T := proc(n,k) return binomial(n,k)*factorial(n-k+1): end: seq(seq(T(n,k),k=0..n),n=0..10); # Nathaniel Johnston, Sep 28 2011

nn=10;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[Exp[y x]/(1-x)^2,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Feb 15 2013 *)

flatten([[binomial(n,k)*factorial(n-k+1) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, May 19 2021

T(n,k) = binomial(n,k)*c(n-k).
From Peter Bala, Jul 10 2008: (Start)
T(n,k) = binomial(n,k)*(n-k+1)!.
T(n,k) = (n-k+1)*T(n-1,k) + T(n-1,k-1).
E.g.f.: exp(x*y)/(1-y)^2 = 1 + (2+x)*y + (6+4*x+x^2)*y^2/2! + ... .
This array is the particular case P(2,1) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below:
  n\k|0....................1...............2.........3.....4
  ----------------------------------------------------------
  0..|1.....................................................
  1..|a....................1................................
  2..|a(a+b)...............2a..............1................
  3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
  4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
  ...
See A094587 for some general properties of these arrays.
Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(1,3) = A136215 and P(2,3) = A136216. (End)
Let f(x) = (1/x^2)*exp(-x). The n-th row polynomial is R(n,x) = (-x)^n/f(x)*(d/dx)^n(f(x)), and satisfies the recurrence equation R(n+1,x) = (x+n+2)*R(n,x)-x*R'(n,x). Cf. A094587. - Peter Bala, Oct 28 2011
Exponential Riordan array [1/(1 - y)^2, y]. The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = Sum_{k=0..n} binomial(n,k)*y^(n-k)*R(k,x). Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = Product_{k = 0..n-1} (2*x + k) with the convention that P(0,x) = 1. Then the present triangle is the triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (24, 18, 6, 1) so P(3,x + 1) = (2*x + 2)*(2*x + 3)*(2*x + 4) = 24 + 18*(2*x) + 6*(2*x)*(2*x + 1) + (2*x)*(2*x + 1)*(2*x + 2). Matrix square of triangle A094587. - Peter Bala, Aug 29 2013
From Tom Copeland, Apr 21 2014: (Start)
T = (I-A132440)^(-2) = {2*I - exp[(A238385-I)]}^(-2) = unsigned exp[2*(I-A238385)] = exp[A005649(.)*(A238385-I)], umbrally, where I = identity matrix.
The e.g.f. is exp(x*y)*(1-y)^(-2), so the row polynomials form an Appell sequence with lowering operator D=d/dx and raising operator x+2/(1-D).
With L(n,m,x) = Laguerre polynomials of order m, the row polynomials are (-1)^n * n! * L(n,-2-n,x) = (-1)^n*(-2!/(-2-n)!)*K(-n,-2-n+1,x) where K is Kummer's confluent hypergeometric function (as a limit of n+s as s tends to zero).
Operationally, (-1)^n*n!*L(n,-2-n,-:xD:) = (-1)^n*x^(n+2)*:Dx:^n*x^(-2-n) = (-1)^n*x^2*:xD:^n*x^(-2) = (-1)^n*n!*binomial(xD-2,n) = (-1)^n*n!*binomial(-2,n)*K(-n,-2-n+1,-:xD:) where :AB:^n = A^n*B^n for any two operators. Cf. A235706.
The generalized Pascal triangle Bala mentions is a special case of the fundamental generalized factorial matrices in A133314. (End)
From Peter Bala, Jul 26 2021: (Start)
O.g.f: 1/y * Sum_{k >= 0} k!*( y/(1 - x*y) )^k =  1 + (2 + x)*y + (6 + 4*x + x^2)*y^2 + ....
First-order recurrence for the row polynomials: (n - x)*R(n,x) = n*(n - x + 1)*R(n-1,x) - x^(n+1) with R(0,x) = 1.
R(n,x) = (x + n + 1)*R(n-1,x) - (n - 1)*x*R(n-2,x) with R(0,x) = 1 and R(1,x) = 2 + x.
R(n,x) = A087981 (x = -2), A000255 (x = -1), A000142 (x = 0), A001339 (x = 1), A081923 (x = 2) and A081924 (x = 3). (End)

Formula 3) in comments corrected by Tom Copeland, Apr 20 2014

Title modified by Tom Copeland, Apr 23 2014

A024000 a(n) = 1 - n.

Original entry on oeis.org

1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30, -31, -32, -33, -34, -35, -36, -37, -38, -39, -40, -41, -42, -43, -44, -45, -46, -47, -48, -49, -50, -51, -52, -53, -54, -55, -56

Offset: 0

N. J. A. Sloane

a(n) is the weighted sum over all derangements (permutations with no fixed points) of n elements where each permutation with an odd number of cycles has weight +1 and each with an even number of cycles has weight -1. [Michael Somos, Jan 19 2011]

			a(4) = -3 because there are 6 derangements with one 4-cycle with weight -1 and 3 derangements with two 2-cycles with weight +1. - _Michael Somos_, Jan 19 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000007, A000166, A087981, A137775. - Michael Somos, Jan 19 2011
Cf. A001489, A094816.
A022958 shifted left.

[1-n: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

A024000:=n->1-n: seq(A024000(n), n=0..100); # Wesley Ivan Hurt, Mar 02 2016

CoefficientList[Series[(1 - 2 x)/(1 - x)^2, {x, 0, 60}], x] Range[0, 60]!
CoefficientList[Series[Exp[x] (1 - x), {x, 0, 60}], x]
1-Range[0,60] (* Harvey P. Dale, Sep 18 2013 *)
Flatten[NestList[(#/.x_/;x>1->Sequence[x,2x])-1&,{1},60]]
(* Robert G. Wilson v, Mar 02 2016 *)

{a(n) = 1 - n} /* Michael Somos, Jan 19 2011 */

E.g.f.: (1-x)*exp(x).
a(n) = Sum_{k=0..n} A094816(n,k)*(-1)^k (alternating row sums of Poisson-Charlier coefficient matrix).
O.g.f.: (1-2*x)/(1-x)^2. a(n+1) = A001489(n). - R. J. Mathar, May 28 2008
a(n) = 2*a(n-1)-a(n-2) for n>1. - Wesley Ivan Hurt, Mar 02 2016

A217924 a(n) = n! * [x^n] exp(2*exp(x) - x - 2). Row sums of triangle A217537.

Original entry on oeis.org

1, 1, 3, 9, 35, 153, 755, 4105, 24323, 155513, 1064851, 7760745, 59895203, 487397849, 4166564147, 37298443977, 348667014723, 3395240969785, 34365336725715, 360837080222761, 3923531021460707, 44108832866004121, 511948390801374835, 6126363766802713481

Offset: 0

Peter Luschny, Oct 15 2012

nonn

The inverse binomial transform of a(n) is A194689.
A087981(n) = Sum_{k=0..n} (-1)^k*s(n+1,k+1)*a(k);
|A000023(n)| = |Sum_{k=0..n} (-1)^(n-k)*s(n,k)*a(k)|
where s(n,k) are the unsigned Stirling numbers of first kind.
a(n) is the number of inequivalent set partitions of {1,2,...,n} where two blocks are considered equivalent when one can be obtained from the other by an alternating (even) permutation. - Geoffrey Critzer, Mar 17 2013

			a(3)=9 because we have: {1,2,3}; {1,3,2}; {1}{2,3}; {1}{3,2}; {2}{1,3}; {2}{3,1}; {3}{1,2}; {3}{2,1}; {1}{2}{3}. - _Geoffrey Critzer_, Mar 17 2013

Vaclav Kotesovec, Table of n, a(n) for n = 0..556

Similar recurrences: A124758, A243499, A284005, A329369, A341392, A372205.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(2*Exp(x) -x-2) ))); // G. C. Greubel, Jan 09 2025

egf := exp(2*exp(x) - x - 2): ser := series(egf, x, 25):
seq(n!*coeff(ser, x, n), n = 0..23);  # Peter Luschny, Apr 22 2024

nn=23;Range[0,nn]!CoefficientList[Series[Exp[2 Exp[x]-x-2],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 17 2013 *)
nmax = 25; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-2*k*x^2 , 1 - (k + 1)*x, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 25 2017 *)

a(n):=sum(sum(binomial(n,k-j)*2^j*(-1)^(k-j)*stirling2(n-k+j,j),j,0,k),k,0,n); /* Vladimir Kruchinin, Feb 28 2015 */

def A217924_list(n):
    T = A217537_triangle(n)
    return [add(T.row(n)) for n in range(n)]
A217924_list(24)

def A217924_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(2*exp(x)-x-2) ).egf_to_ogf().list()
print(A217924_list(40)) # G. C. Greubel, Jan 09 2025

G.f.: 1/Q(0) where Q(k) = 1 + x*k - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013
E.g.f.: exp(2*exp(x) - x - 2). -  Geoffrey Critzer, Mar 17 2013
G.f.: 1/Q(0), where Q(k) = 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: T(0)/(1-x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-x-x*k)*(1-2*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013
a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,k-j)*2^j*(-1)^(k-j)*Stirling2(n-k+j,j). - Vladimir Kruchinin, Feb 28 2015
a(n) = exp(-2) * Sum_{k>=0} 2^k * (k - 1)^n / k!. - Ilya Gutkovskiy, Jun 27 2020
Conjecture: a(n) = Sum_{k=0..2^n-1} A372205(k). - Mikhail Kurkov, Nov 21 2021 [Rewritten by Peter Luschny, Apr 22 2024]
a(n) ~ 2 * n^(n-1) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-1)). - Vaclav Kotesovec, Jun 26 2022

Name extended by a formula of Geoffrey Critzer by Peter Luschny, Apr 22 2024

A137775 Number of triples of permutations on n letters such that for each j, exactly one of the permutations fixes j and the other two have the same image on j.

Original entry on oeis.org

1, 0, 3, 6, 45, 252, 1935, 16146, 153657, 1616760, 18699579, 235498590, 3207570597, 46968796404, 735689606535, 12272343940458, 217191191400945, 4064131571557104, 80166987477918963, 1662468879466624950, 36156426996107254941, 822876672690142595820

Offset: 0

Mark W. Meckes (mark.meckes(AT)case.edu), May 06 2008

nonn

This sequence arises in a calculation of the fourth moments of the volumes of random polytopes in certain very symmetric convex bodies.

			a(2) = 3 because one of the permutations must be the identity and the other two are the transposition (1 2); there are three ways to pick which is the identity.
a(4) = 45 because there are 6 derangements with one 4-cycle with 3^1 ways to color each derangement and 3 derangements with two 2-cycles with 3^2 ways to color each derangement. - _Michael Somos_, Jan 19 2011

M. Meckes, Volumens of symmetric random polytopes, Arch. Math. 82 (2004) 85--96.

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

Cf. A000007, A000166, A024000, A087981.

Range[0, 20]! CoefficientList[Series[Exp[ -3x]/(1 - x)^3, {x, 0, 20}], x]

{a(n) = if( n<0, 0, n! * polcoeff( exp( -3 * x + x * O(x^n) ) / ( 1 - x )^3, n ) )} /* Michael Somos, Jan 19 2011 */

a(n) = (n-1) * (a(n-1) + 3*a(n-2)) with a(0)=1. [corrected by Seiichi Manyama, Apr 23 2025]
E.g.f.: exp(-3x)/(1-x)^3.
a(n) is the number of derangements (permutations with no fixed points) of n elements where each cycle is colored with one of three colors. - Michael Somos, Jan 19 2011
G.f.: hypergeom([1,3],[],x/(1+3*x))/(1+3*x). - Mark van Hoeij, Nov 08 2011
a(n) ~ n! * exp(-3) * n^2/2. - Vaclav Kotesovec, Oct 08 2013
a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * binomial(k+2,2)/(n-k)!. - Seiichi Manyama, Apr 23 2025

Added a(0)=1 by Michael Somos, Jan 19 2011

A052124 Expansion of e.g.f. exp(-2*x)/(1-x)^3.

Original entry on oeis.org

1, 1, 4, 16, 88, 568, 4288, 36832, 354688, 3781504, 44199424, 561823744, 7714272256, 113769309184, 1793341407232, 30085661765632, 535170830467072, 10060645294440448, 199287423535808512, 4148644277780217856, 90545807649965080576, 2067407731760475406336, 49285894020028992323584

Offset: 0

N. J. A. Sloane, Jan 23 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64(b).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A052127, A082031.
Cf. A000023, A087981.
Cf. A000153, A137775.

A052124 := proc(n) option remember; if n <=1 then 1 else n*A052124(n-1)+2*(n-1)*A052124(n-2); fi; end; # Detlef Pauly

Table[(n+5)*(n+2)*n!*Sum[(-1)^k*2^(k+2)*(k+3)/(k+5)!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 28 2012 *)
With[{nn=20},CoefficientList[Series[Exp[(-2x)]/(1-x)^3,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 23 2017 *)

my(x='x+O('x^25)); Vec(serlaplace( exp(-2*x)/(1-x)^3)) \\ Michel Marcus, Oct 25 2021

from math import factorial
from fractions import Fraction
def A052124(n): return int((n+5)*(n+2)*factorial(n)*sum(Fraction((-1 if k&1 else 1)*(k+3)<Chai Wah Wu, Apr 20 2023

a(n) = n*a(n-1) + 2*(n-1)*a(n-2). - Detlef Pauly (dettodet(AT)yahoo.de), Sep 22 2003
a(n) = (n+5)*(n+2)*n! * Sum_{k=0..n} (-1)^k*2^(k+2)*(k+3)/(k+5)!. - Vaclav Kotesovec, Oct 28 2012
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+3)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
a(n) ~ n!*(n+5)*(n+2)/(2*exp(2)). - Vaclav Kotesovec, Jun 15 2013
From Peter Bala, Sep 20 2013: (Start)
a(n) ~ (1/2)*n^2*n!/e^2 for large n.
Letting n -> infinity in the above series for a(n) given by Kotesovec gives the series expansion 1/e^2 = Sum_{k >= 0} (-1)^k*(k+3)*2^(k+3)/(k+5)!.
The sequence b(n) := (1/2)*n!*(n+2)*(n+5) satisfies the recurrence for a(n) given above by Pauly but with the starting values b(0) = 5 and b(1) = 9. This leads to the finite continued fraction expansion a(n) = (1/2)*n!*(n+2)*(n+5)( 1/(5 + 4/(1 + 2/(2 + 4/(3 + ... + 2*(n-1)/n)))) ), valid for n >= 2. Letting n -> infinity in the previous result gives the infinite continued fraction expansion 1/e^2 = 1/(5 + 4/(1 + 2/(2 + 4/(3 + ... + 2*(n-1)/(n + ...))))). Cf. A082031. (End)
a(n) = A087981(n+2)/(2*(n+1)). - Seiichi Manyama, Apr 25 2025

A089006 Number of distinct n X n (0,1) matrices after double sorting: by row, by column, by row .. until reaching a fixed point.

Original entry on oeis.org

1, 2, 7, 45, 650, 24520, 2625117, 836488618, 818230288201, 2513135860300849, 24686082394548211147, 787959836124458000837941, 82905574521614049485027140026

Offset: 0

Wouter Meeussen, Nov 03 2003

Also, number of n X n binary matrices with both rows and columns, considered as binary numbers, in nondecreasing order. (Ordering only rows gives A060690.) - R. H. Hardin, May 08 2008
A result of Adolf Mader and Otto Mutzbauer shows that the two definitions are equivalent. - Victor S. Miller, Feb 03 2009
For n=5, only 0.07% remain distinct. Sorting columns and\or rows does not change the permanent of the matrix and leaves the absolute value of the determinant unchanged.
Diagonal of A180985.

			The 7 (2 X 2)-matrices are {{0,0},{0,0}}, {{0,0},{0,1}}, {{0,0},{1,1}}, {{0,1},{0,1}}, {{0,1},{1,0}}, {{0,1},{1,1}} and {{1,1},{1,1}}.

Adolf Mader and Otto Mutzbauer, "Double Orderings of (0,1) Matrices", Ars Combinatoria v. 61 (2001) pp 81-95.

R. H. Hardin, Binary arrays with both rows and cols sorted, symmetries
M. Werner, An Algorithmic Approach for the Zarankiewicz Problem, Slides, 2012. - From _N. J. A. Sloane_, Jan 01 2013

Cf. A088672, A087981, A180985.

Column 0 of A374525.

baseform[li_List] := FixedPoint[Sort[Transpose[Sort[Transpose[Sort[ #1]]]]]&, li]; Table[Length@Split[Sort[baseform/@(Partition[ #, n]&/@(IntegerDigits[Range[0, -1+2^n^2], 2, n^2]))]], {n, 4}]

a(6)-a(12) found by R. H. Hardin, May 08 2008. These terms were found using bdd's (binary decision diagrams), just setting up the logical relations between bits in a gigantic bdd expression and using that to count the satisfying states.

Edited by N. J. A. Sloane, Feb 05 2009 at the suggestion of Victor S. Miller

A217701 Permanent of the n X n matrix with all diagonal entries n and all off diagonal entries 1.

Original entry on oeis.org

1, 1, 5, 38, 393, 5144, 81445, 1512720, 32237681, 775193984, 20759213061, 612623724800, 19751688891385, 690721009155072, 26039042401938917, 1052645311044368384, 45424010394042998625, 2083972769418997760000, 101288683106200561501189, 5199006109868903819575296

Offset: 0

Jim Pitman, Mar 19 2013

nonn

a(n) is the number of terms that appear before cancellations occur in the evaluation of the determinant of an n X n matrix by the usual sum over permutations of [n], for a matrix whose off diagonal entries are specified, and each of whose diagonal entries is minus the sum of the negatives of the off diagonal entries and one additional term, as in the usual presentation of the matrix in the Matrix Tree Theorem.
The number a(n-1) - n^(n-2) (A000272) is the number of terms which cancel in Zeilberger's proof of the Matrix Tree Theorem. This number is even, as the terms which cancel are equal in magnitude with opposite sign, and as is also apparent from the formula in terms of A087981(n) which is a corollary of Zeilberger's proof.
Formula involves the derangement numbers A000166 which count permutations with no fixed points, also the number A087981 of colored permutations with no fixed points of n elements where each cycle is one of two colors.
Number of permutations of [n] where the fixed points are n-colored and all other points are unicolored. - Alois P. Heinz, Apr 23 2020

Alois P. Heinz, Table of n, a(n) for n = 0..386
Doron Zeilberger, A combinatorial approach to matrix algebra, Discrete Mathematics, 56 (1985), 61-72.

Also closely related to A058127.
Main diagonal of A089258.
Cf. A176043.

a:= n-> n!*coeff(series(exp((n-1)*x)/(1-x), x, n+1), x, n):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 23 2020
# second Maple program:
b:= proc(n, k) option remember; `if`(n<1, 1-n,
      (n+k-1)*b(n-1, k)+(k-1)*(1-n)*b(n-2, k))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 23 2020
# third Maple program:
b:= proc(n, k) option remember;
      `if`(min(n, k)<0, 0, n*b(n-1, k)+(k-1)^n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 23 2020

derange[0,z_]:=1; derange[n_,z_]:= Pochhammer[z,n] - Sum[ Binomial[n,k] z^(n-k) derange[k,z], {k,0,n-1}]; a[n_]:= Sum[ Binomial[n,k] derange[n-k,1] n^k, {k,0,n}] ; a/@ Range[0,10]
derange[0,z_]:=1; derange[n_,z_]:= Pochhammer[z,n] - Sum[ Binomial[n,k] z^(n-k) derange[k,z], {k,0,n-1}]; a[n_]:= Sum[ Binomial[n,j] derange[n-j,2] (n+1)^(j-1) (n-j+1), {j,0,n}]; a/@ Range[0,10]
(* Alternative: *)
a[n_] := Exp[n - 1] Gamma[n + 1, n - 1];
Table[a[n], {n, 0, 19}]  (* Peter Luschny, Dec 24 2021 *)

a(n) = Sum_{k=0..n} C(n,k)*D_{n-k}*n^k, where D_n = A000166(n).
a(n) = Sum_{j=0..n} C(n,k)*D_{n-k,2} (n+1)^(j-1) (n-j+1) where D_{n,2} = A087981(n).
a(n) = n! * [x^n] exp((k-1)*x)/(1-x). - Alois P. Heinz, Apr 23 2020
a(n) = exp(n-1)*Gamma(n+1, n-1). - Peter Luschny, Dec 24 2021

a(0),a(16)-a(19) from Alois P. Heinz, Apr 23 2020

A295181 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x)^k.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 3, 4, 9, 0, 1, 0, 4, 6, 24, 44, 0, 1, 0, 5, 8, 45, 128, 265, 0, 1, 0, 6, 10, 72, 252, 880, 1854, 0, 1, 0, 7, 12, 105, 416, 1935, 6816, 14833, 0, 1, 0, 8, 14, 144, 620, 3520, 16146, 60032, 133496, 0, 1, 0, 9, 16, 189, 864, 5725, 31104, 153657, 589312, 1334961, 0

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

A(n,k) is the k-fold exponential convolution of A000166 with themselves, evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + k*x^2/2! + 2*k*x^3/3! + 3*k*(k + 2)*x^4/4! + 4*k*(5*k + 6)*x^5/5! + 5*k*(3*k^2 + 26*k + 24)*x^6/6! + ...
Square array begins:
  1,   1,    1,    1,    1,    1,  ...
  0,   0,    0,    0,    0,    0,  ...
  0,   1,    2,    3,    4,    5,  ...
  0,   2,    4,    6,    8,   10,  ...
  0,   9,   24,   45,   72,  105,  ...
  0,  44,  128,  252,  416,  620,  ...

N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Columns k=0..5 give A000007, A000166, A087981, A137775, A383344, A383384.
Rows n=0..3 give A000012, A000004, A001477, A005843.
Main diagonal gives A295182.
Cf. A008279, A265609.

Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

a(n, k) = n!*sum(j=0, n, (-k)^(n-j)*binomial(j+k-1, j)/(n-j)!); \\ Seiichi Manyama, Apr 25 2025

E.g.f. of column k: exp(-k*x)/(1 - x)^k.
From Seiichi Manyama, Apr 25 2025: (Start)
A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k-1,j)/(n-j)!.
A(0,k) = 1, A(1,k) = 0; A(n,k) = (n-1) * (A(n-1,k) + k*A(n-2,k)). (End)

cp's OEIS Frontend

A010027 Triangle read by rows: T(n,k) is the number of permutations of [n] having k consecutive ascending pairs (0 <= k <= n-1).

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A000023 Expansion of e.g.f. exp(-2*x)/(1-x).

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A132159 Lower triangular matrix T(n,j) for double application of an iterated mixed order Laguerre transform inverse to A132014. Coefficients of Laguerre polynomials (-1)^n * n! * L(n,-2-n,x).

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A024000 a(n) = 1 - n.

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A217924 a(n) = n! * [x^n] exp(2*exp(x) - x - 2). Row sums of triangle A217537.

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