A158882 -id:A158882 - cp's OEIS frontend

A133942 a(n) = (-1)^n * n!.

1, -1, 2, -6, 24, -120, 720, -5040, 40320, -362880, 3628800, -39916800, 479001600, -6227020800, 87178291200, -1307674368000, 20922789888000, -355687428096000, 6402373705728000, -121645100408832000, 2432902008176640000

Offset: 0

Michael Somos, Sep 30 2007

A variant of A000142, the factorial numbers. - N. J. A. Sloane, Oct 03 2007
The terms of this sequences form the factorial series which Euler called the divergent series par excellence.
Euler summed this series to 0.596347... (A073003 = Gompertz's constant).
Sum_{n>=0} 1/a(n) = 1/e. - Jaume Oliver Lafont, Mar 03 2009
A002104(n+1) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Apr 30 2012
a(n) = A048594(2*n+1, n+1). - Reinhard Zumkeller, Mar 02 2014
log(1+x) = Sum_{n>=1} a(n-1)/n!*x^n. - James R. Buddenhagen, May 24 2015
It seems that a(n) is the determinant of n+1 X n+1 matrix whose elements are m(i,j) = quotient(i/j) + remainder(i/j). - Andres Cicuttin, Feb 11 2018

			G.f. = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 - 120*x^5 + 720*x^6 - 5040*x^7 + ...

A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p. 141 (10.19)
R. Roy, Sources in the Development of Mathematics, Cambridge University Press, 2011. See p. 186.

Iain Fox, Table of n, a(n) for n = 0..449 (first 161 terms from Vincenzo Librandi)
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.
V. S. Varadarajan, Euler and his Work on Infinite Series, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 515-539. (See p. 527 and 530.)
Index entries for sequences related to factorial numbers

Partial sums are A058006.
Alternating row sums of A048994.
Also, a(n) = A048994(n+1,1).
Cf. A000142, A002104, A002741, A055209, A158882.

List([0..20],n->(-1)^n*Factorial(n)); # Muniru A Asiru, Oct 27 2018

a133942 n = a133942_list !! n
a133942_list = zipWith (*) a000142_list $ cycle [1, -1]
-- Reinhard Zumkeller, Mar 02 2014

[(-1)^n * Factorial(n): n in [0..25]]; // Vincenzo Librandi, May 12 2011

seq((-1)^n*factorial(n),n=0..20); # Muniru A Asiru, Oct 27 2018

nn=20;CoefficientList[Series[1/(1+x),{x,0,nn}],x]Range[0,nn]! (* or *)
RecurrenceTable[{a[0]==1,a[n]==-n*a[n-1]},a[n],{n,20}] (* Harvey P. Dale, May 10 2011 and slightly modified by Robert G. Wilson v, Feb 12 2018 *)
a[n_] := (-1)^n*n!; Array[a, 22, 0] (* Robert G. Wilson v, Feb 11 2018 *)
Times@@@Partition[Riffle[Range[0,30]!,{1,-1}],2] (* Harvey P. Dale, Dec 30 2019 *)

PARI
```
{a(n) = if( n<0, 0, (-1)^n * n! )};
```

import math
for n in range(0, 25): print((-1)**n*math.factorial(n), end=', ') # Stefano Spezia, Oct 27 2018

Sum_{i=0..n} (-1)^i * i^n * binomial(n, i) = (-1)^n * n!. - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
Stirling transform of a(n) = [1, -1, 2, -6, 24, ...] is A000007(n) = [1, 0, 0, 0, 0, ...].
a(n) = -n * a(n-1) unless n=0. a(n) = (-1)^n * A000142(n).
E.g.f.: 1/(1 + x).
G.f.: integral(t=1/x,infinity, (e^-t)/t) e^(1/x)/x = 1/(1 + x/(1 + x/(1 + 2*x/(1 + 2*x/(1 + 3*x/(1 + 3*x/(1 + ...))))))).
Convolution inverse of A158882. HANKEL transform is A055209. PSUM transform is A058006. BIN1 transform is A002741(n+1). - Michael Somos, Apr 30 2012
G.f.: 1 - x/(G(0)+x) where G(k) = 1 + (k+1)*x/(1 + x*(k+2)/G(k+1)), G(0) = W(1,1;x)/W(1,2;x), W(a,b,x) = 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)]; (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 14 2012
G.f.: 1/U(0) where U(k) = 1 + x*(k+1)/(1 + x*(k+1)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 15 2012
a(n) = (-1)^n*det(S(i+1,j)|, 1 <= i,j <= n), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 06 2013
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(k+1)/(2*x*(k+1) + 1 + 2*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
E.g.f.: 1/(1 + x)= G(0), where G(k) = 1 - x*(k+1)*(k+2)/(1 + (k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 29 2014
For n >= 1, a(n) = round(zeta^(n)(2)), where zeta^(n) is the n-th derivative of the Riemann zeta function. - Iain Fox, Nov 13 2017
a(n) = (n+1)^(n+1) * Integral_{x=0..1} (x*log(x))^n dx. - Peter James Foreman, Oct 27 2018

A167894 Expansion of g.f.: 1/(Sum_{k >= 0} k!*x^k).

Original entry on oeis.org

1, -1, -1, -3, -13, -71, -461, -3447, -29093, -273343, -2829325, -31998903, -392743957, -5201061455, -73943424413, -1123596277863, -18176728317413, -311951144828863, -5661698774848621, -108355864447215063

Offset: 0

Philippe Deléham, Nov 15 2009

sign

Essentially the same as A003319, which is the main entry for these numbers. - N. J. A. Sloane, Jun 11 2013

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 40.

G. C. Greubel, Table of n, a(n) for n = 0..400

Cf. A003319, A158882.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( 1/(&+[Factorial(k)*x^k: k in [0..m+1]]) )); // G. C. Greubel, Feb 07 2019

CoefficientList[Series[1/(Sum[k!*x^k, {k, 0, 25}]), {x, 0, 20}], x] (* G. C. Greubel, Jun 30 2016 *)

a(n) := if n=0 then 1 else -sum( a(i)*(n-i)!,i,0,n-1); /* Vladimir Kruchinin, Oct 10 2024 */

m=20; my(x='x+O('x^m)); Vec(1/sum(k=0,m+1, k!*x^k)) \\ G. C. Greubel, Feb 07 2019

def A167894_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] * k
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A167894_list(20)) # Peter Luschny, Feb 19 2016

m=20; (1/sum(factorial(k)*x^k for k in range(m+1))).series(x, m).coefficients(x, sparse=False) # G. C. Greubel, Feb 07 2019

a(n) = - Sum_{i=0..n-1} a(i)*(n-i)! for n > 0 with a(0) = 1. - Vladimir Kruchinin, Oct 10 2024
From Sergei N. Gladkovskii, Jun 24 2012, Oct 15 2012, Nov 18 2012, Dec 26 2012, Apr 25 2013, May 29 2013, Aug 08 2013, Nov 19 2013: (Start) Continued fractions:
G.f.: 1 - x/Q(0), where Q(k) = 1 - (k+1)*x/(1 - (k+2)*x/Q(k+1)).
G.f.: U(0) where U(k) =  1 - x*(k+1)/(1 - x*(k+1)/U(k+1)).
G.f.: 1/G(0) where G(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))).
G.f.: A(x) = 1 - x/G(0) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1).
G.f.: x*Q(0), where Q(k) = 1/x - 1 - 2*k - (k+1)^2/Q(k+1).
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))).
G.f.: 2/Q(0), where Q(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/Q(k+1) )).
G.f.: conjecture: Q(0), where Q(k) = 1 + k*x - (k+1)*x/Q(k+1). (End)
a(n) ~ -n! * (1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - 25912/n^7 - 319339/n^8 - 4388949/n^9 - 66495386/n^10 - ...). - Vaclav Kotesovec, Dec 08 2020

A158883 G.f. satisfies: [x^n] A(x)^(n+1) = [x^n] A(x)^n for n>1 with A(0)=A'(0)=1.

Original entry on oeis.org

1, 1, -2, 9, -56, 425, -3726, 36652, -397440, 4695489, -59941550, 821711605, -12037503384, 187689245588, -3104186515976, 54295661153700, -1001685184237056, 19444296845046033, -396260414466644574, 8460628832978195683, -188898511962856879400

Offset: 0

Paul D. Hanna, Apr 30 2009

sign

			G.f.: A(x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 + ...
(d/dx) (x/A(x)) = 1 - 2*x + 9*x^2 - 56*x^3 + 425*x^4 - 3726*x^5 + ...
1/A(x) = 1 - x + 3*x^2 - 14*x^3 + 85*x^4 + ... + (-1)^n*A088716(n)*x^n + ...
where a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1.
...
Coefficients of powers of g.f. A(x) begin:
A^1: 1,1,-2,9,-56,425,-3726,36652,-397440,4695489,...;
A^2: 1,2,(-3),14,-90,702,-6297,63144,-695886,8334822,...;
A^3: 1,3,(-3),(16),-108,870,-7997,81774,-915798,11116902,...;
A^4: 1,4,-2,(16),(-115),960,-9050,94368,-1073658,13204560,...;
A^5: 1,5,0,15,(-115),(996),-9630,102365,-1182690,14730890,...;
A^6: 1,6,3,14,-111,(996),(-9870),106890,-1253466,15804548,...;
A^7: 1,7,7,14,-105,973,(-9870),(108816),-1294412,16514162,...;
A^8: 1,8,12,16,-98,936,-9704,(108816),(-1312227),16931984,...;
A^9: 1,9,18,21,-90,891,-9426,107406,(-1312227),(17116900),...;
A^10:1,10,25,30,-80,842,-9075,104980,-1298625,(17116900),...; ...
where coefficients [x^n] A(x)^(n+1) and [x^n] A(x)^n are
enclosed in parenthesis and equal (n+1)*A158884(n) for n > 1:
[ -3,16,-115,996,-9870,108816,-1312227,17116900,...];
compare to A158884:
[1,1,-1,4,-23,166,-1410,13602,-145803,1711690,-21785618,...]
and also to the logarithmic derivative of A158884:
[1,-3,16,-115,996,-9870,108816,-1312227,17116900,...].

Alois P. Heinz, Table of n, a(n) for n = 0..448

Cf. A088716, A158884, variant: A158882.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(j)*b(n-j-1)*(j+1), j=0..n-1))
    end:
a:= n-> `if`(n=0, 1, -(-1)^n*n*b(n-1)):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 18 2020

m = 19; A[_] = 1;
Do[A[x_] = 1 + x*D[x/A[x], x] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Feb 18 2020 *)

Composita(n,k,F):=if k=1 then F(n) else sum(F(i+1)*Composita(n-i-1,k-1,F),i,0,n-k);
array(a, 10);
a[1]:1;
af(n):=a[n];
for n:2 thru 10 do a[n]:n*sum(Composita(n-1, k, af)*(-1)^k , k, 1, n-1);
makelist(af(n),n,1,10); /* Vladimir Kruchinin, Dec 01 2011 */

{a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0);A[ #A]=(Vec(Ser(A)^(#A-1))-Vec(Ser(A)^(#A)))[ #A]);A[n+1]}

G.f. satisfies: A(x) = 1 + x*(d/dx)(x/A(x)) so that x^2*A'(x) = x*A(x) + A(x)^2 - A(x)^3.
a(n) = (-1)^(n-1)*n*A088716(n-1) for n >= 1.
G.f.: A(x) = 1/(Sum_{n>=0} (-1)^n*A088716(n)*x^n), where g.f. F(x) of A088716 satisfies: F(x) = 1 + x*F(x)*(d/dx)(x*F(x)).
G.f. satisfies: [x^n] A(x)^(n+1) = (n+1)*A158884(n) for n > 1.

A208896 Rectangular table where the g.f. of row n satisfies: R(n,x) = 1 + xR(n,x)^n [d/dx x/R(n,x)] for n>=0, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, -2, 1, 1, -1, 9, 1, 1, 0, 3, -56, 1, 1, 1, 0, -13, 425, 1, 1, 2, 0, 0, 71, -3726, 1, 1, 3, 3, -1, 0, -461, 36652, 1, 1, 4, 9, 0, 1, 0, 3447, -397440, 1, 1, 5, 18, 19, -12, 0, 0, -29093, 4695489, 1, 1, 6, 30, 72, 0, -14, 0, 0, 273343, -59941550

Offset: 0

Paul D. Hanna, Mar 03 2012

The following property accounts for the zeros along the main diagonal.
The row g.f.s satisfy: [x^k] R(n,x)^(k-n+1) = [x^k] R(n,x)^(k-n) for k>=2
and thus when k=n we have [x^n] R(n,x) = [x^n] R(n,x)^0 = 0 for n>=2.

			Coefficients in the n-th row g.f., R(n,x), of this table begins:
n=0: [1, 1,-2,  9, -56,  425, -3726,  36652, -397440, 4695489, ...];
n=1: [1, 1,-1,  3, -13,   71,  -461,   3447,  -29093,  273343, ...];
n=2: [1, 1, 0,  0,   0,    0,     0,      0,       0,       0, ...];
n=3: [1, 1, 1,  0,  -1,    1,     0,      0,      -5,      27, ...];
n=4: [1, 1, 2,  3,   0,  -12,   -14,     43,      96,     -50, ...];
n=5: [1, 1, 3,  9,  19,    0,  -195,   -732,    -453,    6495, ...];
n=6: [1, 1, 4, 18,  72,  201,     0,  -4200,  -27984,  -91044, ...];
n=7: [1, 1, 5, 30, 175,  880,  3106,      0, -114485,-1124735, ...];
n=8: [1, 1, 6, 45, 344, 2451, 14946,  64522,       0,-3805692, ...];
n=9: [1, 1, 7, 63, 595, 5453, 45927, 331177, 1704795,       0, ...];
n=10:[1, 1, 8, 84, 944,10550,112336,1094604, 9157984,55095601, 0,...]; ...
in which the main diagonal is zeros for n>=2.
Initial row g.f.s are illustrated by the following.
R(0,x) = 1 + x*[d/dx x/R(0,x)] begins:
R(0,x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 +...
which satisfies: [x^k] R(0,x)^(k+1) = [x^k] R(0,x)^k for k>=2.
...
R(1,x) = 1 + x*R(1,x)*[d/dx x/R(1,x)] begins:
R(1,x) = 1 + x - x^2 + 3*x^3 - 13*x^4 + 71*x^5 - 461*x^6 + 3447*x^7 +...
which satisfies: [x^k] R(1,x)^k = [x^k] R(1,x)^(k-1) for k>=2.
...
R(2,x) = 1 + x*R(2,x)^2*[d/dx x/R(2,x)] is satisfied by:
R(2,x) = 1 + x,
which satisfies: [x^k] R(2,x)^(k-1) = [x^k] R(2,x)^(k-2) = 0 for k>=2.
...
R(3,x) = 1 + x*R(3,x)^3*[d/dx x/R(3,x)] begins:
R(3,x) = 1 + x + x^2 - x^4 + x^5 - 5*x^8 + 27*x^9 - 147*x^10 + 996*x^11 +...
which satisfies: [x^k] R(3,x)^(k-2) = [x^k] R(3,x)^(k-3) for k>=2.
...

Paul D. Hanna, Rows n = 0..46, flattened.

Cf. A158883 (row 0), A158882 (row 1), A208897, A208898.

{T(n,k)=local(ROWn=1+x+x*O(x^k));for(i=0,k,ROWn=1+x*ROWn^n*deriv(x/ROWn));polcoeff(ROWn,k)}
for(n=0,12,for(k=0,12,print1(T(n,k),","));print(""))

A237652 G.f. satisfies: [x^n] A(x)^(n^2) = [x^n] A(x)^(n^2-1) for n>1 with A(0)=A'(0)=1.

Original entry on oeis.org

1, 1, -3, 20, -245, 4290, -114422, 4086800, -203647509, 12920587070, -1053926397590, 105178069321944, -12765014959365682, 1838898931467398164, -311221726754896488780, 61047560951879121055296, -13747598006865584455353165, 3521759025274977423306328182, -1018406456608128511401443183654

Offset: 0

Paul D. Hanna, May 07 2014

sign

			G.f.: A(x) = 1 + x - 3*x^2 + 20*x^3 - 245*x^4 + 4290*x^5 - 114422*x^6 +...
The coefficients in relevant powers of g.f. A(x)  begin:
A^3: [1, 3, (-6), 43, -597, 11127, -313038, 11486268, ...];
A^4: [1, 4, (-6), 48, -721, 13836, -399342, 14835168, ...];
...
A^8: [1, 8,   4, (48), -1022, 21328, -677040, 26240352, ...];
A^9: [1, 9,   9, (48), -1071, 22572, -732768, 28655712, ...];
...
A^15: [1, 15, 60, 125, (-1260), 26508,  -986720, 40214775, ...];
A^16: [1, 16, 72, 160, (-1260), 26688, -1018704, 41720576, ...];
...
A^24: [1, 24, 204,  848,  54, (25680), -1211936, 50397024, ...];
A^25: [1, 25, 225, 1000, 525, (25680), -1230900, 51117200, ...];
...
A^35: [1, 35, 490, 3675, 14035, 52927, (-1360590), 54736260, ...];
A^36: [1, 36, 522, 4080, 16695, 61452, (-1360590), 54781344, ...];
...
A^48: [1, 48,  984, 11488, 82428, 399936, -450096, (53190144), ...];
A^49: [1, 49, 1029, 12348, 91679, 460110, -217266, (53190144), ...];
...
which illustrates [x^n] A(x)^(n^2-1) = [x^n] A(x)^(n^2) for n>1.

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A158882, A171791.

{a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[ #A]=(Vec(Ser(A)^((#A-1)^2-1))-Vec(Ser(A)^((#A-1)^2)))[ #A]); A[n+1]}
for(n=0,30,print1(a(n),", "))

A306576 Expansion of 1/(1 - x - 2x/(1 - 2x - 3x/(1 - 3x - 4x/(1 - 4x - 5*x/(1 - ...))))), a continued fraction.

Original entry on oeis.org

1, 3, 19, 179, 2183, 32355, 562343, 11198203, 251297263, 6275390067, 172639089031, 5189033793611, 169220733646271, 5951777459480931, 224604052936701815, 9053124776482735291, 388198017158108201839, 17645733672934447166163, 847577245047341210277415

Offset: 0

Ilya Gutkovskiy, Jun 25 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..386

Cf. A000311, A001147, A003319, A158882.

$RecursionLimit = Infinity; nmax = 18; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-k x, 1 - k x, {k, 2, nmax + 1}]), {x, 0, nmax}], x]

a(n) ~ c * d^n * n^(n+1), where d = 1 / (exp(1) * (2*log(2) - 1)) = 0.952329306865721945... and c = 1/(sqrt(2) * (2*log(2) - 1)^(3/2)) = 2.945150206105358... - Vaclav Kotesovec, Jul 01 2019, updated May 06 2024

cp's OEIS Frontend

A133942 a(n) = (-1)^n * n!.

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A167894 Expansion of g.f.: 1/(Sum_{k >= 0} k!*x^k).

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A158883 G.f. satisfies: [x^n] A(x)^(n+1) = [x^n] A(x)^n for n>1 with A(0)=A'(0)=1.

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A208896 Rectangular table where the g.f. of row n satisfies: R(n,x) = 1 + x*R(n,x)^n * [d/dx x/R(n,x)] for n>=0, as read by antidiagonals.

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A237652 G.f. satisfies: [x^n] A(x)^(n^2) = [x^n] A(x)^(n^2-1) for n>1 with A(0)=A'(0)=1.

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A306576 Expansion of 1/(1 - x - 2*x/(1 - 2*x - 3*x/(1 - 3*x - 4*x/(1 - 4*x - 5*x/(1 - ...))))), a continued fraction.

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A208896 Rectangular table where the g.f. of row n satisfies: R(n,x) = 1 + xR(n,x)^n [d/dx x/R(n,x)] for n>=0, as read by antidiagonals.

A306576 Expansion of 1/(1 - x - 2x/(1 - 2x - 3x/(1 - 3x - 4x/(1 - 4x - 5*x/(1 - ...))))), a continued fraction.