A136580 -id:A136580 - cp's OEIS frontend

A003422 Left factorials: !n = Sum_{k=0..n-1} k!.

0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114, 4037914, 43954714, 522956314, 6749977114, 93928268314, 1401602636314, 22324392524314, 378011820620314, 6780385526348314, 128425485935180314, 2561327494111820314, 53652269665821260314, 1177652997443428940314

Offset: 0

N. J. A. Sloane, R. K. Guy

Number of {12, 12*, 1*2, 21*}- and {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
a(n) is the number of permutations on [n] that avoid the patterns 2n1 and n12. An occurrence of a 2n1 pattern is a (scattered) subsequence a-n-b with a > b. - David Callan, Nov 29 2007
Also, numbers left over after the following sieving process: At step 1, keep all numbers of the set N = {0, 1, 2, ...}. In step 2, keep only every second number after a(2) = 2: N' = {0, 1, 2, 4, 6, 8, 10, ...}. In step 3, keep every third of the numbers following a(3) = 4, N" = {0, 1, 2, 4, 10, 16, 22, ...}. In step 4, keep every fourth of the numbers beyond a(4) = 10: {0, 1, 2, 4, 10, 34, 58, ...}, and so on. - M. F. Hasler, Oct 28 2010
If s(n) is a second-order recurrence defined as s(0) = x, s(1) = y, s(n) = n*(s(n - 1) - s(n - 2)), n > 1, then s(n) = n*y - n*a(n - 1)*x. - Gary Detlefs, May 27 2012
a(n) is the number of lists of {1, ..., n} with (1st element) = (smallest element) and (k-th element) <> (k-th smallest element) for k > 1, where a list means an ordered subset. a(4) = 10 because we have the lists: [1], [2], [3], [4], [1, 3, 2], [1, 4, 2], [1, 4, 3], [2, 4, 3], [1, 3, 4, 2], [1, 4, 2, 3]. Cf. A000262. - Geoffrey Critzer, Oct 04 2012
Consider a tree graph with 1 vertex. Add an edge to it with another vertex. Now add 2 edges with vertices to this vertex, and then 3 edges to each open vertex of the tree (not the first one!), and the next stage is to add 4 edges, and so on. The total number of vertices at each stage give this sequence (see example). - Jon Perry, Jan 27 2013
Additive version of the superfactorials A000178. - Jon Perry, Feb 09 2013
Repunits in the factorial number system (see links). - Jon Perry, Feb 17 2013
Whether n|a(n) only for 1 and 2 remains an open problem. A published 2004 proof was retracted in 2011. This is sometimes known as Kurepa's conjecture. - Robert G. Wilson v, Jun 15 2013, corrected by Jeppe Stig Nielsen, Nov 07 2015.
!n is not always squarefree for n > 3. Miodrag Zivkovic found that 54503^2 divides !26541. - Arkadiusz Wesolowski, Nov 20 2013
a(n) gives the position of A007489(n) in A227157. - Antti Karttunen, Nov 29 2013
Matches the total domination number of the Bruhat graph from n = 2 to at least n = 5. - Eric W. Weisstein, Jan 11 2019
For the connection with Kurepa trees, see A. Petojevic, The {K_i(z)}{i=1..oo} functions, Rocky Mtn. J. Math., 36 (2006), 1637-1650. - _Aleksandar Petojevic, Jun 29 2018
This sequence converges in the p-adic topology, for every prime number p. - Harry Richman, Aug 13 2024

			!5 = 0! + 1! + 2! + 3! + 4! = 1 + 1 + 2 + 6 + 24 = 34.
x + 2*x^2 + 4*x^3 + 10*x^4 + 34*x^5 + 154*x^6 + 874*x^7 + 5914*x^8 + 46234*x^9 + ...
From _Arkadiusz Wesolowski_, Aug 06 2012: (Start)
Illustration of initial terms:
.
. o        o         o            o                         o
.          o         o            o                         o
.                   o o          o o                       o o
.                              ooo ooo                   ooo ooo
.                                             oooo oooo oooo oooo oooo oooo
.
. 1        2         4            10                        34
.
(End)
The tree graph. The total number of vertices at each stage is 1, 2, 4, 10, ...
    0 0
    |/
    0-0
   /
0-0
   \
    0-0
    |\
    0 0
- _Jon Perry_, Jan 27 2013

Richard K. Guy, Unsolved Problems Number Theory, Section B44.
D. Kurepa, On the left factorial function !n. Math. Balkanica 1 1971 147-153.
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 = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
Bernd C. Kellner, Some remarks on Kurepa's left factorial, arXiv:math/0410477 [math.NT], 2004.
D. Kurepa, On the left factorial function !N, Math. Balkanica 1 (1971), 147-153. (Annotated scanned copy).
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Romeo Meštrović, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013.
Romeo Meštrović, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.
Hisanori Mishima, Factorizations of many number sequences.
Hisanori Mishima, Factorizations of many number sequences
F. J. Papp, Letter to N. J. A. Sloane, Nov 1974.
Jon Perry, Sum of Factorials. [Broken link?]
Aleksandar Petojevic, On Kurepa's hypothesis for the left factorial, Filomat, Vol. 12, No. 1, 1998.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Aleksandar Petojevic, The {K_i(z)}_{i=1..oo} functions, Rocky Mtn. J. Math., 36 (2006), 1637-1650.
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Left Factorial.
Eric Weisstein's World of Mathematics, Repunit.
Wikipedia, Factorial number system.
Miodrag Zivkovic, The number of primes sum_{i=1..n} (-1)^(n-i)*i! is finite, Math. Comp. 68 (1999), pp. 403-409.
Index entries for sequences related to factorial numbers.

Equals A007489(n-1)+1 for n>=1. Cf. A000142, A014144, A005165.
Twice A014288. See also A049782, A100612.
Cf. A102639, A102411, A102412, A101752, A094216, A094638, A008276, A000166, A000110, A000204, A000045, A000108, A135722, A227157, A000178, A137338, A232845, A357145.

a003422 n = a003422_list !! n
a003422_list = scanl (+) 0 a000142_list
-- Reinhard Zumkeller, Dec 27 2011

A003422 := proc(n) local k; add(k!,k=0..n-1); end proc:
# Alternative, using the Charlier polynomials A137338:
C := proc(n, x) option remember; if n > 0 then (x-n)*C(n-1, x) - n*C(n-2, x)
elif n = 0 then 1 else 0 fi end: A003422 := n -> (-1)^(n+1)*C(n-1, -1):
seq(A003422(n), n=0..22); # Peter Luschny, Nov 28 2018
# third Maple program:
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+(n-1)!) end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 24 2022

Table[Sum[i!, {i, 0, n - 1}], {n, 0, 20}] (* Stefan Steinerberger, Mar 31 2006 *)
Join[{0}, Accumulate[Range[0, 25]!]] (* Harvey P. Dale, Nov 19 2011 *)
a[0] = 0; a[1] = 1; a[n_] := a[n] = n*a[n - 1] - (n - 1)*a[n - 2]; Array[a, 23, 0] (* Robert G. Wilson v, Jun 15 2013 *)
a[n_] := (-1)^n*n!*Subfactorial[-n-1]-Subfactorial[-1]; Table[a[n] // FullSimplify, {n, 0, 22}] (* Jean-François Alcover, Jan 09 2014 *)
RecurrenceTable[{a[n] == n a[n - 1] - (n - 1) a[n - 2], a[0] == 0, a[1] == 1}, a, {n, 0, 10}] (* Eric W. Weisstein, Jan 11 2019 *)
Range[0, 20]! CoefficientList[Series[(ExpIntegralEi[1] - ExpIntegralEi[1 - x]) Exp[x - 1], {x, 0, 20}], x] (* Eric W. Weisstein, Jan 11 2019 *)
Table[(-1)^n n! Subfactorial[-n - 1] - Subfactorial[-1], {n, 0, 20}] // FullSimplify (* Eric W. Weisstein, Jan 11 2019 *)
Table[(I Pi + ExpIntegralEi[1] + (-1)^n n! Gamma[-n, -1])/E, {n, 0, 20}] // FullSimplify (* Eric W. Weisstein, Jan 11 2019 *)

makelist(sum(k!,k,0,n-1), n, 0, 20); /* Stefano Spezia, Jan 11 2019 */

a003422(n)=sum(k=0,n-1,k!) \\ Charles R Greathouse IV, Jun 15 2011

from itertools import count, islice
def A003422_gen(): # generator of terms
    yield from (0,1)
    c, f = 1, 1
    for n in count(1):
        yield (c:= c + (f:= f*n))
A003422_list = list(islice(A003422_gen(),20)) # Chai Wah Wu, Jun 22 2022

def a(n):
    if n == 0: return 0
    s = f = 1
    for k in range(1, n):
        f *= k
        s += f
    return round(s)
print([a(n) for n in range(24)])  # Peter Luschny, Mar 05 2024

D-finite with recurrence: a(n) = n*a(n - 1) - (n - 1)*a(n - 2). - Henry Bottomley, Feb 28 2001
Sequence is given by 1 + 1*(1 + 2*(1 + 3*(1 + 4*(1 + ..., terminating in n*(1)...). - Jon Perry, Jun 01 2004
a(n) = Sum_{k=0..n-1} P(n, k) / C(n, k). - Ross La Haye, Sep 20 2004
E.g.f.: (Ei(1) - Ei(1 - x))*exp(-1 + x) where Ei(x) is the exponential integral. - Djurdje Cvijovic and Aleksandar Petojevic, Apr 11 2000
a(n) = Integral_{x = 0..oo} [(x^n - 1)/(x - 1)]*exp(-x) dx. - Gerald McGarvey, Oct 12 2007
A007489(n) = !(n + 1) - 1 = a(n + 1) - 1. - Artur Jasinski, Nov 08 2007. Typos corrected by Antti Karttunen, Nov 29 2013
Starting (1, 2, 4, 10, 34, 154, ...), = row sums of triangle A135722. - Gary W. Adamson, Nov 25 2007
a(n) = a(n - 1) + (n - 1)! for n >= 2. - Jaroslav Krizek, Jun 16 2009
E.g.f. A(x) satisfies the differential equation A'(x) = A(x) + 1/(1 - x). - Vladimir Kruchinin, Jan 19 2011
a(n + 1) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A182386(k) for k = 0, 1, ..., n. - Michael Somos, Apr 27 2012
From Sergei N. Gladkovskii, May 09 2013 to Oct 22 2013: (Start)
Continued fractions:
G.f.: x/(1-x)*Q(0) where Q(k) = 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/Q(k+1))).
G.f.: G(0)*x/(1-x)/2 where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))).
G.f.: 2*x/(1-x)/G(0) where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))).
G.f.: W(0)*x/(1+sqrt(x))/(1-x) where W(k) = 1 + sqrt(x)/(1 - sqrt(x)*(k+1)/(sqrt(x)*(k+1) + 1/W(k+1))).
G.f.: B(x)*(1+x)/(1-x) where B(x) is the g.f. of A153229.
G.f.: x/(1-x) + x^2/(1-x)/Q(0) where Q(k) = 1 - 2*x*(2*k+1) - x^2*(2*k+1)*(2*k+2)/(1 - 2*x*(2*k+2) - x^2*(2*k+2)*(2*k+3)/Q(k+1)).
G.f.: x*(1+x)*B(x) where B(x) is the g.f. of A136580. (End)
a(n) = (-1)^(n+1)*C(n-1, -1) where C(n, x) are the Charlier polynomials (with parameter a=1) as given in A137338. (Evaluation at x = 1 gives A232845.) - Peter Luschny, Nov 28 2018
a(n) = (a(n-3)*(n-2)^2*(n-3)! + a(n-1)^2)/a(n-2) (empirical). - Gary Detlefs, Feb 25 2022
a(n) = signum(n)/b(1,n) with b(i,n) = i - [iMohammed Bouras, Sep 07 2022
Sum_{n>=1} 1/a(n) = A357145. - Amiram Eldar, Oct 01 2022

A058006 Alternating factorials: 0! - 1! + 2! - ... + (-1)^n n!

Original entry on oeis.org

1, 0, 2, -4, 20, -100, 620, -4420, 35900, -326980, 3301820, -36614980, 442386620, -5784634180, 81393657020, -1226280710980, 19696509177020, -335990918918980, 6066382786809020, -115578717622022980, 2317323290554617020, -48773618881154822980

Offset: 0

Henry Bottomley, Nov 13 2000

From Harry Richman, Aug 13 2024: (Start)
Euler argued this sequence converges to 0.596347... (A073003 = Gompertz's constant); see Lagarias Section 2.5.
This sequence converges in the p-adic topology, for every prime number p. (End)

			a(5) = 0!-1!+2!-3!+4!-5! = 1-1+2-6+24-120 = -100.
G.f. = 1 + 2*x^2 - 4*x^3 + 20*x^4 - 100*x^5 + 620*x^6 - 4420*x^7 + 35900*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
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.
Eric Weisstein's MathWorld, Incomplete Gamma Function.

Cf. A000142, A003422, A005165, A153229 (absolute values), A136580.

Partial sums of A133942.

a058006 n = a058006_list !! n
a058006_list = scanl1 (+) a133942_list
-- Reinhard Zumkeller, Mar 02 2014

a[ n_] := Sum[ (-1)^k k!, {k, 0, n}]; (* Michael Somos, Jan 28 2014 *)

{a(n) = sum(k=0, n, (-1)^k * k!)}; /* Michael Somos, Jan 28 2014 */

a(n) = (-1)^n n! + a(n-1) = A005165(n)(-1)^n + 1.
a(n) = -(n-1)*a(n-1) + n*a(n-2), n>0.
E.g.f.: d/dx ((GAMMA(0,1)-GAMMA(0,1+x))*exp(1+x)). - Max Alekseyev, Jul 05 2010
G.f.: G(0)/(1-x), where G(k)= 1 - (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
0 = a(n)*(-a(n+1) + a(n+3)) + a(n+1)*(2*a(n+1) - 2*a(n+2) -a(n+3)) + a(n+2)*(a(n+2)) if n>=-1. - Michael Somos, Jan 28 2014
a(n) = exp(1)*Gamma(0,1) + (-1)^n*exp(1)*(n+1)!*Gamma(-n-1,1), where Gamma(a,x) is the upper incomplete Gamma function. - Vladimir Reshetnikov, Oct 29 2015

Corrections and more information from Michael Somos, Feb 19 2003

A153229 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).

Original entry on oeis.org

0, 1, 0, 2, 4, 20, 100, 620, 4420, 35900, 326980, 3301820, 36614980, 442386620, 5784634180, 81393657020, 1226280710980, 19696509177020, 335990918918980, 6066382786809020, 115578717622022980, 2317323290554617020, 48773618881154822980, 1075227108896452857020

Offset: 0

Shaojun Ying (dolphinysj(AT)gmail.com), Dec 21 2008

Previous name was: Weighted Fibonacci numbers.
From Peter Bala, Aug 18 2013: (Start)
The sequence occurs in the evaluation of the integral I(n) := Integral_{u >= 0} exp(-u)*u^n/(1 + u) du.
The result is I(n) = A153229(n) + (-1)^n*I(0), where I(0) = Integral_{u >= 0} exp(-u)/(1 + u) du = 0.5963473623... is known as Gompertz's constant. See A073003.
Note also that I(n) = n!*Integral_{u >= 0} exp(-u)/(1 + u)^(n+1) du. (End)
((-1)^(n+1))*a(n) = p(n,-1), where the polynomials p are defined at A248664. - Clark Kimberling, Oct 11 2014

			a(20) = 19 * a(18) + 18 * a(19) = 19 * 335990918918980 + 18 * 6066382786809020 = 6383827459460620 + 109194890162562360 = 115578717622022980

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

Cf. A000045, A000255, A000587, A058006.
First differences of A136580.
Column k=0 of A303697 (for n>0).

unsigned long a(unsigned int n) {
if (n == 0) return 0;
if (n == 1) return 1;
return (n - 1) * a(n - 2) + (n - 2) * a(n - 1); }

t1 := sum(n!*x^n, n=0..100): F := series(t1/(1+x), x, 100): for i from 0 to 40 do printf(`%d, `, i!-coeff(F, x, i)) od: # Zerinvary Lajos, Mar 22 2009
# second Maple program:
a:= proc(n) a(n):= `if`(n<2, n, (n-1)*a(n-2) +(n-2)*a(n-1)) end:
seq(a(n), n=0..25); # Alois P. Heinz, May 24 2013

Join[{a = 0}, Table[b = n! - a; a = b, {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==(n-1)a[n-2]+(n-2)a[n-1]},a,{n,30}] (* Harvey P. Dale, May 01 2020 *)

a(n)=if(n,my(t=(-1)^n);-t-sum(i=1,n-1,t*=-i),0); \\ Charles R Greathouse IV, Jun 28 2011

a(0) = 0, a(1) = 1, and for n >= 2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).
For n>=1, a(n) = A058006(n-1) * (-1)^(n-1).
G.f.: G(0)*x/(1+x)/2, where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: 2*x/(1+x)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: W(0)*x/(1+sqrt(x))/(1+x), where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+1)/(sqrt(x)*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 17 2013
a(n) ~ (n-1)! * (1 - 1/n + 1/n^3 + 1/n^4 - 2/n^5 - 9/n^6 - 9/n^7 + 50/n^8 + 267/n^9 + 413/n^10), where numerators are Rao Uppuluri-Carpenter numbers, see A000587. - Vaclav Kotesovec, Mar 16 2015
E.g.f.: exp(1)/exp(x)*(Ei(1, 1-x)-Ei(1, 1)). - Alois P. Heinz, Jul 05 2018
a(n) = Sum_{k = 0..n-1} (-1)^(n-k-1) * k!. - Peter Bala, Dec 05 2024

Edited by Max Alekseyev, Jul 05 2010

Better name by Joerg Arndt, Aug 17 2013

A358499 a(n) = Sum_{k=0..floor(n/4)} (n-4*k)!.

Original entry on oeis.org

1, 1, 2, 6, 25, 121, 722, 5046, 40345, 363001, 3629522, 39921846, 479041945, 6227383801, 87181920722, 1307714289846, 20923268929945, 355693655479801, 6402460887648722, 121646408123121846, 2432922931445569945, 51091297865364919801, 1124007130238495328722

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A136580, A358498, A358500.

PARI
```
a(n) = sum(k=0, n\4, (n-4*k)!);
```

a(n) = n * a(n-1) + a(n-4) - n * a(n-5) for n > 4.

a(n) ~ n! * (1 + 1/n^4 + 6/n^5 + 25/n^6 + 90/n^7 + 302/n^8 + 994/n^9 + 3487/n^10 + ...), for coefficients see A099948. - Vaclav Kotesovec, Nov 24 2022

A358498 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)!.

Original entry on oeis.org

1, 1, 2, 7, 25, 122, 727, 5065, 40442, 363607, 3633865, 39957242, 479365207, 6230654665, 87218248442, 1308153733207, 20929020542665, 355774646344442, 6403681859461207, 121666029429374665, 2433257782822984442, 51097345853568901207, 1124122393807037054665

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A136580, A358499, A358500.

PARI
```
a(n) = sum(k=0, n\3, (n-3*k)!);
```

a(n) = n * a(n-1) + a(n-3) - n * a(n-4) for n > 3.

a(n) ~ n! * (1 + 1/n^3 + 3/n^4 + 7/n^5 + 16/n^6 + 46/n^7 + 203/n^8 + 1178/n^9 + 7242/n^10 + ...), for coefficients see A143817. - Vaclav Kotesovec, Nov 24 2022

A358500 a(n) = Sum_{k=0..floor(n/5)} (n-5*k)!.

Original entry on oeis.org

1, 1, 2, 6, 24, 121, 721, 5042, 40326, 362904, 3628921, 39917521, 479006642, 6227061126, 87178654104, 1307677996921, 20922829805521, 355687907102642, 6402379932789126, 121645187587486104, 2432903315854636921, 51090963094539245521, 1124001083465514782642

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A136580, A358498, A358499.

PARI
```
a(n) = sum(k=0, n\5, (n-5*k)!);
```

a(n) = n * a(n-1) + a(n-5) - n * a(n-6) for n > 5.

a(n) ~ n! * (1 + 1/n^5 + 10/n^6 + 65/n^7 + 350/n^8 + 1701/n^9 + 7771/n^10 + 34150/n^11 + 146905/n^12 + ...), the coefficients are Sum_{j=0..(k-4)/5} Stirling2(k,5*j+4). - Vaclav Kotesovec, Nov 24 2022

A358607 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (n-2*k)!.

Original entry on oeis.org

1, 1, 1, 5, 23, 115, 697, 4925, 39623, 357955, 3589177, 39558845, 475412423, 6187461955, 86702878777, 1301486906045, 20836087009223, 354385941189955, 6381537618718777, 121290714467642045, 2426520470557921223, 50969651457241797955, 1121574207307049758777

Offset: 0

Seiichi Manyama, Nov 23 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A358608, A358609, A358611.

Cf. A121868, A136580.

PARI
```
a(n) = sum(k=0, n\2, (-1)^k*(n-2*k)!);
```

a(n) = n * a(n-1) - a(n-2) + n * a(n-3) for n > 2.
a(n) ~ n! * (1 - 1/n^2 - 1/n^3 + 5/n^5 + 23/n^6 + 74/n^7 + 161/n^8 - 57/n^9 - 3466/n^10 - ...), for coefficients see A121868. - Vaclav Kotesovec, Nov 25 2022
a(2n) = 1+A215096(2n). a(2n+1) = A215096(2n+1). - R. J. Mathar, Jun 14 2024

A136579 Triangle read by rows: A128174 * A136572.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 6, 1, 0, 2, 0, 24, 0, 1, 0, 6, 0, 120, 1, 0, 2, 0, 24, 0, 720, 0, 1, 0, 6, 0, 120, 0, 5040, 1, 0, 2, 0, 24, 0, 720, 0, 40320

Offset: 0

Gary W. Adamson, Jan 09 2008

Row sums = A136580: 1, 1, 3, 7, 27, 127, ...

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 2;
  0, 1, 0, 6;
  1, 0, 2, 0, 24;
  0, 1, 0, 6,  0, 120;
  1, 0, 2, 0, 24,   0, 720;
  ...

Cf. A136580, A128174, A136572, A000142.

A128174 * A136572 Triangle, even rows = even n! interspersed with zeros. Odd n rows, = odd n! interspersed with zeros.

T(2*i,2*k) = (2*k)! = A010050(k). T(2*i+1,2*k+1) = (2*k+1)! = A009445(k). - R. J. Mathar, Jun 04 2021

A173279 Irregular triangle read by rows: M(n,k) = (n-2*k)!, k=0..floor(n/2).

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 24, 2, 1, 120, 6, 1, 720, 24, 2, 1, 5040, 120, 6, 1, 40320, 720, 24, 2, 1, 362880, 5040, 120, 6, 1, 3628800, 40320, 720, 24, 2, 1, 39916800, 362880, 5040, 120, 6, 1, 479001600, 3628800, 40320, 720, 24, 2, 1, 6227020800, 39916800, 362880, 5040, 120, 6, 1, 87178291200

Offset: 0

Gary W. Adamson, Feb 14 2010

In the limit as j-> infinity, the power M^j approaches the limit described in A173280.

Row sums: sum_{k=0..n/2} M(n,k) = A136580(n).

			Triangle starts in row n=0 as:
1;
1;
2, 1;
6, 1;
24, 2, 1;
120, 6, 1;
720, 24, 2, 1;
5040, 120, 6, 1;
40320, 720, 24, 2, 1;
362880, 5040, 120, 6, 1;
3628800, 40320, 720, 24, 2, 1;
39916800, 362880, 5040, 120, 6, 1;
479001600, 3628800, 40320, 720, 24, 2, 1;
...

Cf. A000142, A136580, A173280

A173279 := proc(n,k) factorial(n-2*k) ; end proc: seq(seq(A173279(n,k),k=0..floor(n/2)),n=0..20) ; # R. J. Mathar, Feb 22 2010

keyword tabl replaced by tabf, R. J. Mathar, Feb 22 2010

A341900 Partial sums of A005165.

Original entry on oeis.org

0, 1, 2, 7, 26, 127, 746, 5167, 41066, 368047, 3669866, 40284847, 482671466, 6267305647, 87660962666, 1313941673647, 21010450850666, 357001369769647, 6423384156578666, 122002101778601647, 2439325392333218666, 51212944273488041647, 1126440053169940898666

Offset: 0

R. J. Mathar, Jun 04 2021

Cf. A000142, A005165, A136580, A215096.

a(n)-a(n-1) = A005165(n).
a(n) = Sum_{i=0..floor((n-1)/2)} (n-2*i)!.
a(n) = A136580(n) - A059841(n).
D-finite with recurrence a(n) -n*a(n-1) -a(n-2) +n*a(n-3)=0.
a(n) = n! + a(n-2) for n >= 2. - Alois P. Heinz, Jun 04 2021

cp's OEIS Frontend

A003422 Left factorials: !n = Sum_{k=0..n-1} k!.

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A058006 Alternating factorials: 0! - 1! + 2! - ... + (-1)^n n!

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A153229 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).

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A358499 a(n) = Sum_{k=0..floor(n/4)} (n-4*k)!.

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A358498 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)!.

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A358500 a(n) = Sum_{k=0..floor(n/5)} (n-5*k)!.

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A358607 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (n-2*k)!.

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