A137775 -id:A137775 - cp's OEIS frontend

A087981 E.g.f.: exp(-2*x) / (1-x)^2.

1, 0, 2, 4, 24, 128, 880, 6816, 60032, 589312, 6384384, 75630080, 972387328, 13483769856, 200571078656, 3185540657152, 53800242216960, 962741176500224, 18195808235880448, 362183230599856128, 7572922094360723456, 165945771111208714240, 3802923921298533384192, 90965940197460917878784, 2267151124921333646884864

Offset: 0

Gordon F. Royle, Oct 28 2003

Permanent of an (n+1) X (n+1) (+1, -1)-matrix with exactly n -1's on the diagonal and 1's everywhere else.
It is conjectured by Kräuter and Seifter that for n >= 5 a(n-1) is the maximal possible value for the permanent of a nonsingular n X n (+1, -1)-matrix. I do not know for which values of n this has been confirmed - compare A087982. - N. J. A. Sloane
The Kräuter conjecture on permanents is true (see Budrevich and Guterman). - Sergei Shteiner, Jan 17 2020
The maximal possible value for the permanent of a singular n X n (+1, -1)-matrix is obviously n!.
Degree of the "hyperdeterminant" of a multilinear polynomial on (\P^1(\C))^n, or equivalently of an element of (\C^2)^{⊗ n}: see Gelfand, Kapranov and Zelevinsky. - Eric Rains, Mar 15 2004
(-1)^n * a(n) = Polynomials in A010027 evaluated at -1. - Ralf Stephan, Dec 15 2004
a(n) is the number of n X n (-1, 0, 1)-matrices containing in every row and every column exactly one -1 and one 1 such that the main diagonal does not contain 0's. - Vladimir Shevelev, Apr 01 2010
a(n) is the number of colored permutations with no fixed points of n elements where each cycle is one of two colors. - Michael Somos, Jan 19 2011
Binomial transform is A000255. Hankel transform is A059332. - Paul Barry, Apr 11 2011
Exponential self-convolution of subfactorials (A000166). - Vladimir Reshetnikov, Oct 07 2016

			G.f. = 1 + 2*x^2 + 4*x^3 + 24*x^4 + 128*x^5 + 880*x^6 + 6816*x^7 + ...
Since a(1) = 0, then, for n = 2, we have a(2) = -(-2)^3/4 = 2; further, for n = 3, we find a(3) = (3*6/5)*2 - (-2)^4/5 = 36/5 - 16/5 = 4. - _Vladimir Shevelev_, Apr 01 2010
a(4) = 24 because there are 6 derangements with one 4-cycle with 2^1 ways to color each derangement and 3 derangements with two 2-cycles with 2^2 ways to color each derangement. - _Michael Somos_, Jan 19 2011

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 1994; see Corollary 2.10 in Chapter 14 (p. 457).

T. D. Noe, Table of n, a(n) for n = 0..100
Mikhail V. Budrevich and Alexander E. Guterman, Kräuter conjecture on permanents is true, arXiv:1810.04439 [math.CO], 2018.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Hyperdeterminants, Advances in Mathematics 96(2) (1992), 226-263; see Corollary 3.10 (p. 246).
Arnold R. Kräuter, Permanenten - Ein kurzer Überblick, Séminaire Lotharingien de Combinatoire, B09b (1983), 34 pp.
Arnold R. Kräuter and Norbert Seifter, Some properties of the permanent of (1,-1)-matrices, Linear and Multilinear Algebra 15 (1984), 207-223.
Norbert Seifter, Upper bounds for permanents of (1,-1)-matrices, Israel J. Math. 48 (1984), 69-78.
Edward Tzu-Hsia Wang, On permanents of (1,-1)-matrices, Israel J. Math. 18 (1974), 353-361.
Wikipedia, Hyperdeterminant
Index entries for sequences related to binary matrices

Cf. A087982, A087983, A176097.

Cf. A000007, A000023, A000166, A024000, A137775. - Michael Somos, Jan 19 2011

seq(simplify(KummerU(-n, -n-1, -2)), n = 0..24); # Peter Luschny, May 10 2022

Range[0, 20]! CoefficientList[Series[Exp[-2 x]/(1 - x)^2, {x, 0, 20}], x]
Table[(-2)^n HypergeometricPFQ[{2, -n}, {}, 1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( -2 * x + x * O(x^n) ) / ( 1 - x )^2, n ) )} /* Michael Somos, Jan 19 2011 */

Krauter and Seifter prove that the permanent of an n X n {-1, 1} matrix is divisible by 2^{n - [log_2(n)] - 1}.
Let c(n) be the permanent of the {-1, +1}-matrix of order n X n with n diagonal -1's only. Let a(n) be the permanent of the {-1, +1}-matrix of order (n+1) X (n+1) with n diagonal -1's only. Then by expanding along the first row (like determinant, but with no sign) we get c(n+1) = -c(n) + n a(n-1), a(n) = c(n) + n a(n-1), with c(2) = 2, a(2) = 2. {c(n)} has e.g.f. exp(-2x)/(1-x), see A000023. Also a(n) = c(n+1) + 2*c(n).
The following 4 formulas hold: a(n) = Sum_{k = 0..n} C(n, k)*D_k*D_{n-k}, where D_n = A000166(n); a(n) = n!*Sum_{j = 0..n} (n+1-j)*(-2)^j/j!; a(0) = 1, a(1) = 0 and, for n > 0, a(n+1) = n*(a(n) + 2*a(n-1)); a(0) = 1 and, for n > 0, a(n) = (n*(n+3)/(n+2))*a(n-1) - (-2)^(n+1)/(n+2). - Vladimir Shevelev, Apr 01 2010 [edited by Michael Somos, Jan 19 2011]
G.f.: 1/(1-2x^2/(1-2x-6x^2/(1-4x-12x^2/(1-6x-20x^2/(1-.../(1-2n*x-(n+1)(n+2)x^2/(1-... (continued fraction). - Paul Barry, Apr 11 2011
E.g.f.: 1/U(0) where U(k)=  1 - 2*x/( 1 + x/(2 - x - 4/( 2 + x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Oct 28 2012
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: 1/Q(0) where Q(k) = 1 - 2*k*x - x^2*(k + 1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 10 2013
G.f.: S(x)/x - 1/x = G(0)/x - 1/x, where S(x) = sum(k >= 0, k!*(x/(1+2*x))^k  ), G(k) = 1 + (2*k + 1)*x/( 1+2*x - 2*x*(1+2*x)*(k+1)/(2*x*(k+1) + (1+2*x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) = (-2)^n*hypergeom([2, -n], [], 1/2) = 4*(-2)^n*(1 - 2*hypergeom([1, -n-3], [], 1/2))/(n^2+3*n+2) = (4*(-2)^n + Gamma(n+4, -2)*exp(-2))/(n^2+3*n+2). - Vladimir Reshetnikov, Oct 07 2016
a(n) ~ sqrt(2*Pi) * n^(n+3/2) / exp(n+2). - Vaclav Kotesovec, Oct 08 2016
a(n) = KummerU(-n, -n - 1, -2). - Peter Luschny, May 10 2022

More terms from Jaap Spies, Oct 28 2003
Further terms from Gordon F. Royle, Oct 29 2003
Definition via e.g.f. from Eric Rains, Mar 15 2004
Changed the offset and terms to correspond to e.g.f, Michael Somos, Jan 19 2011

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

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

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)

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

Original entry on oeis.org

1, 0, 2, 6, 72, 620, 8640, 122346, 2156672, 41367672, 905126400, 21646532270, 570077595648, 16268377195044, 502096929431552, 16629319748711250, 588938142209310720, 22196966267762213744, 887352465220427317248, 37496112562144553167062, 1670071417348195942400000, 78195398849926292810318940

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

nonn

The n-th term of the n-fold exponential convolution of A000166 with themselves.

Robert Israel, Table of n, a(n) for n = 0..396
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Main diagonal of A295181.

Cf. A000166, A000407, A001813, A087981, A137775, A295183, A383379.

S:= series((exp(-x)/(1-x))^n,x,30):
seq(n!*coeff(S,x,n),n=0..29); # Robert Israel, Nov 16 2017

Table[n! SeriesCoefficient[Exp[-n x]/(1 - x)^n, {x, 0, n}], {n, 0, 21}]

a(n) = A295181(n,n).
a(n) ~ phi^(3*n - 1/2) * n^n / (5^(1/4) * exp(n*(1 + 1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017
a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k-1,k)/(n-k)!. - Seiichi Manyama, Apr 25 2025

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A087981 E.g.f.: exp(-2*x) / (1-x)^2.

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A024000 a(n) = 1 - n.

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

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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.

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A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

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A383344 Expansion of e.g.f. exp(-4*x) / (1-x)^4.

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A383382 Expansion of e.g.f. exp(-3*x) / (1-x)^5.

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