A000522 -id:A000522 - cp's OEIS frontend

A038154 a(n) = n! * Sum_{k=0..n-2} 1/k!.

0, 0, 2, 12, 60, 320, 1950, 13692, 109592, 986400, 9864090, 108505100, 1302061332, 16926797472, 236975164790, 3554627472060, 56874039553200, 966858672404672, 17403456103284402, 330665665962403980, 6613313319248079980, 138879579704209680000

Offset: 0

N. J. A. Sloane

nonn

The number of rank-orderings of (>=2)-element subsets of an n-set. (Counts nontrivial votes in a rank-ordering voting system.) E.g., a(5) = 320 = 120+120+60+20 because of 5-, 4-, 3- and 2-element subsets. - Warren D. Smith, Jul 06 2005
a(n) is the number of simple cycles through a vertex of the complete graph K_(n+1) on n+1 vertices [Hassani]. For example, in the complete graph K_4 with vertex set {A,B,C,D} there are a(3) = 12 simple cycles at the vertex A, namely the six 3-cycles ABCA, ABDA, ACBA, ACDA, ADBA and ADCA and the six 4-cycles ABCDA, ABDCA, ACBDA, ACDBA, ADBCA and ADCBA. The sum of the lengths of the cycles at a vertex of K_n is equal to A141834(n). - Peter Bala, Jul 09 2008
See A000522 for the number of paths between a pair of distinct vertices of K_n. - Peter Bala, Jul 09 2008
a(n) = n*a(n-1) + A000217(n-1), where A000217(n) is the n-th triangular number. - Gary Detlefs, May 20 2010

			0=1*0+0, 2=2*0+2, 12=3*2+6, 60=4*12+12, 320 = 5*60+20, ... - _Gary Detlefs_, May 20 2010

Mehdi Hassani, Counting and computing by e, arXiv:math/0606613 [math.CO], 2006.
J. Sawada and A. Williams, Successor rules for flipping pancakes and burnt pancakes, Preprint 2015.
Index entries for sequences related to factorial numbers

Cf. A000522, A038155, A141834.

Table[n!Sum[1/k!,{k,0,n-2}],{n,0,30}] (* Harvey P. Dale, Jun 04 2012 *)

main(size)=my(k); vector(size,n,(n-1)!*sum(k=0,n-3,1/k!)); \\ Anders Hellström, Jul 14 2015

a(n) = A007526(n) - n.
a(n) = floor(n!*exp(1))-n-1, n>0. - Vladeta Jovovic, Aug 25 2001
a(n) = n*a(n-1) + n*(n-1), for n>=3, a(2)=2 and a(3) = 12. - Ian Myers, Jul 19 2012
a(n) = A000522(n-2) * n*(n-1). - Doug Bell, Jun 30 2015
E.g.f.: exp(x)*x^2/(1 - x). - Ilya Gutkovskiy, Jan 26 2017
a(n) = 2*A038155(n). - Alois P. Heinz, Jan 26 2017

A046662 Sum of mistyped version of binomial coefficients.

Original entry on oeis.org

1, 2, 7, 52, 749, 17686, 614227, 29354312, 1844279257, 147273109354, 14561325802271, 1745720380045852, 249461639720702917, 41886684733511640062, 8164388189339113521259, 1828191138807263097870256, 466057478369217965809683377, 134193343258948416556377786322

Offset: 0

Len Smiley

Origin of the name of this sequence: Binomial coefficients are n!/((n-k)!*k!) but if parentheses are omitted in the denominator, the formula might result in n!/(n-k)!*k! = n!*k!/(n-k)! and the sum giving a(n) instead of 2^n. If k! is forgotten altogether, one gets A000522. - Olivier Gérard, Mar 05 2024

Binomial transform of (n!)^2. - Peter Luschny, May 31 2014

Seiichi Manyama, Table of n, a(n) for n = 0..253
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013

Cf. A003149, A064570, A229464.

Cf. A000522 (Total number of ordered k-subsets of [1,n], k=0..n.)

Table[Sum[(n!k!)/(n-k)!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 29 2012 *)

a(n) = Sum_{k=0..n} n!*k!/(n-k)!.
E.g.f.: exp(x)*F(x), with F(x) = Sum_{k>=0} k!*x^k. - Ralf Stephan, Apr 02 2004
a(n) = n^2*a(n - 1) - n*(n - 1)*a(n - 2) + 1. - Vladeta Jovovic, Jul 15 2004
From Peter Bala, Nov 26 2017: (Start)
a(k) == a(0) (mod k) for all k (by the inhomogeneous recurrence equation).
More generally, a(n+k) = a(n) (mod k) for all n and k (by an induction argument on n). It follows that for each positive integer k, the sequence a(n) (mod k) is periodic, with the exact period dividing k. For example, modulo 10 the sequence becomes 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, 1, 2, 7, 2, 9, 6, 7, 2, 7, 4, ... with exact period 10. (End)
G.f.: Sum_{k>=0} (k!)^2*x^k/(1 - x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019
a(n) ~ (n!)^2. - Vaclav Kotesovec, May 03 2021
a(n) = 3F0(1,1,-n;;-1). - R. J. Mathar, Jun 26 2024

Corrected and extended by Harvey P. Dale, Sep 29 2012

A054091 Row sums of A054090.

Original entry on oeis.org

1, 2, 4, 10, 32, 130, 652, 3914, 27400, 219202, 1972820, 19728202, 217010224, 2604122690, 33853594972, 473950329610, 7109254944152, 113748079106434, 1933717344809380, 34806912206568842, 661331331924808000, 13226626638496160002, 277759159408419360044

Offset: 0

Clark Kimberling

nonn

Row sums of A094816 as a triangular array as follows: {1}, {1,1}, {1,3}, {1,1,8}, {6,1,1,24}, {29,10,1,1,89}, ... - Michael Somos, Nov 19 2006
a(n) = (n-1)a(n-1)+2, n>0; 2=0*1+2, 4=1*2+2, 10=2*4+2, ... - Gary Detlefs, May 20 2010
Row sums of triangle A208058. - Gary W. Adamson, Feb 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings, arXiv:2501.00357 [math.CO], 2024. See p. 15.

Cf. A000522, A054090, A094816, A208058.

[n le 2 select n else (n-1)*Self(n-1) -(n-3)*Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 23 2022

a:= n-> `if`( n=0, 1, add(2*(n-1)!/j!, j=0..n-1)): seq(a(n), n=0..18); # Zerinvary Lajos, Oct 20 2006
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, 2+(n-1)*a(n-1))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 23 2022

Table[If[n==0, 1, 2*(n-1)!*Sum[1/j!, {j,0,n-1}]], {n,0,30}] (* G. C. Greubel, Jun 23 2022 *)

{a(n)= local(A); if(n<1, n==0, A=vector(n); A[1]=2; for(k=1, n-1, A[k+1]=k*A[k]+2); A[n])} /* Michael Somos, Nov 19 2006 */

{a(n)= if(n<1, n==0, n--; n!*polcoeff( 2*exp(x+x*O(x^n))/(1-x), n))} /* Michael Somos, Nov 19 2006 */

[1]+[2*factorial(n-1)*sum(1/factorial(j) for j in (0..n-1)) for n in (1..30)] # G. C. Greubel, Jun 23 2022

a(n+1) = 2*A000522(n).
a(n+1) = Sum(2*n!/j!, j=0..n). - Zerinvary Lajos, Oct 20 2006
a(n) = 2*floor(e*(n-1)!), n>1. - Gary Detlefs, May 20 2010
a(n) = n*a(n-1) - (n-2)*a(n-2), a(0)=1, a(1)=2. - Vincenzo Librandi, Feb 23 2012

A064383 Integers n >= 1 such that n divides 0!-1!+2!-3!+4!-...+(-1)^{n-1}(n-1)!.

Original entry on oeis.org

1, 2, 4, 5, 10, 13, 20, 26, 37, 52, 65, 74, 130, 148, 185, 260, 370, 463, 481, 740, 926, 962, 1852, 1924, 2315, 2405, 4630, 4810, 6019, 9260, 9620, 12038, 17131, 24076, 30095, 34262, 60190, 68524, 85655, 120380, 171310, 222703, 342620, 445406, 890812, 1113515

Offset: 1

Kevin Buzzard (buzzard(AT)ic.ac.uk), Sep 28 2001

If a is in the sequence, then so are all its positive divisors. If a and b are coprime and in the sequence, then so is their product. Hence in extending the sequence, one may as well just look for primes in the sequence (and then check powers of these primes). Heuristically one might expect a very sparse but infinite set of primes in the sequence, but the largest one I know is p=463 and I've searched up to 600000. This sequence was brought to my attention by David Loeffler.
Also, n such that A000522(n)==1 (mod n^2). - Benoit Cloitre, Apr 15 2003
The primes in this sequence are the same as the terms > 1 in A124779. - Jonathan Sondow, Nov 09 2006
Also, n such that n|A(n-1), where A(0) = 1 and A(k) = k*A(k-1)+1 = A000522(k) for k > 0. - Jonathan Sondow, Dec 22 2006
Michael Mossinghoff has calculated that 2, 5, 13, 37, 463 are the only primes in the sequence up to 150 million. - Jonathan Sondow, Jun 12 2007

			4 is in the sequence because 4 divides 0!-1!+2!-3!=1-1+2-6=-4.

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer-Verlag, 2004, B43.

J. Sondow, The Taylor series for e and the primes 2, 5, 13, 37, 463: a surprising connection
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Index entries for sequences related to factorial numbers

Cf. A000522, A057245, A064384, A124779, A129924.

s = 0; Do[ s = s + (-1)^(n)(n)!; If[ Mod[ s, n + 1 ] == 0, Print[ n + 1 ] ], {n, 0, 600000} ]
Divisors[4454060] (* From Formula above *) (* Harvey P. Dale, Aug 09 2012 *)

Up to n=600000, these are just the divisors of 4*5*13*37*463.

More terms from Sean A. Irvine, Jul 02 2023

A080954 E.g.f. exp(5x)/(1-x).

Original entry on oeis.org

1, 6, 37, 236, 1569, 10970, 81445, 648240, 5576545, 52142030, 531185925, 5891873300, 70946620225, 923526766050, 12935478240325, 194062691183000, 3105155646818625, 52788408935369750, 950195175533921125

Offset: 0

Paul Barry, Feb 26 2003

Binomial transform of A053487. 4th Binomial transform of A000522. Fifth binomial transform of n! = A000142.

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

Cf. A008290, A010842, A053486.

F(x):= exp(5*x)/(1-x): f[0]:=F(x): for n from 1 to 20 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n], n=0..18); # Zerinvary Lajos, Apr 03 2009

With[{nn=20},CoefficientList[Series[Exp[5x]/(1-x),{x,0,nn}],x] Range[0, nn]!] (* Harvey P. Dale, Sep 19 2011 *)
Table[n!*Sum[5^k/k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 28 2012 *)
RecurrenceTable[{a[0]==1, a[n]==n*a[n-1] + 5^n}, a, {n, 20}] (* Vincenzo Librandi, Nov 15 2012 *)

a(n) = n! * Sum_{k=0..n} 5^k/k!.
a(n) is the permanent of the n X n matrix with 6's on the diagonal and 1's elsewhere. a(n) = Sum_{k=0..n} A008290(n, k)*6^k. - Philippe Deléham, Dec 12 2003
Conjecture: -a(n) + (n+5)*a(n-1) + 5*(1-n)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ n!*exp(5). - Vaclav Kotesovec, Oct 13 2012
a(0)=0, a(n) = n * a(n-1) + 5^n. - Vincenzo Librandi, Nov 15 2012

A096341 E.g.f.: exp(x)/(1-x)^7.

Original entry on oeis.org

1, 8, 71, 694, 7421, 86276, 1084483, 14665106, 212385209, 3280842496, 53862855551, 936722974958, 17205245113141, 332864226563324, 6766480571358971, 144202473398010826, 3215159679583864433

Offset: 0

Philippe Deléham, Jun 28 2004

nonn

Sum_{k=0..n} A094816(n,k)*x^k give A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n) for x = 1, 2, 3, 4, 5, 6 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. E.g.f. exp(x)/(1-x)^k: A000522 (k = 1), A001339 (k = 2), A082030 (k = 3), A095000 (k = 4), A095177 (k = 5), A096307 (k = 6).

Cf. A094795, A094816.

Table[HypergeometricPFQ[{7, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)
With[{nn = 250}, CoefficientList[Series[Exp[x]/(1 - x)^7, {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*7^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+6)!/6!.
a(n) = 2F0(7,-n;;-1). - Benedict W. J. Irwin, May 27 2016
From Peter Bala, Jul 26 2021: (Start)
a(n) = (n+7)*a(n-1) - (n-1)*a(n-2) with a(0) = 1 and a(1) = 8.
First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) + 1 with a(0) = 1, where P(n) = n^6 + 15*n^5 + 100*n^4 + 355*n^3 + 694*n^2 + 689*n + 265 = A094795(n).
(End)

A123901 a(n) = (n+3)/gcd(d(n), d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0..n} 1/k! to lowest terms.

Original entry on oeis.org

3, 4, 5, 3, 7, 4, 9, 1, 11, 6, 13, 7, 3, 8, 17, 9, 19, 2, 21, 11, 23, 12, 5, 1, 27, 14, 29, 3, 31, 16, 33, 17, 7, 18, 37, 19, 3, 4, 41, 21, 43, 22, 9, 23, 47, 24, 49, 5, 51, 2, 53, 27, 11, 28, 57, 29, 59, 6, 61, 31, 63, 32, 1, 33, 67, 34, 69, 7, 71, 36, 73, 1, 15, 38, 77, 3, 79, 8, 81

Offset: 0

Jonathan Sondow, Oct 18 2006

			a(5) = 4 because (5+3)/gcd(d(5),d(7)) = 8/gcd(2,20) = 8/2 = 4.

Antti Karttunen, Table of n, a(n) for n = 0..4096
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, arXiv:0709.0671 [math.NT], 2007-2009; Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

Cf. A000522, A061354, A093101, A123899, A123900, A124781.

(A[n_] := If[n==0,1,n*A[n-1]+1]; d[n_] := GCD[A[n],n! ]; Table[(n+3)/GCD[d[n],d[n+2]], {n,0,79}])
(* Second program, faster: *)
Table[(n + 3)/Apply[GCD, Map[GCD[#!, Floor[E*#!] - Boole[# == 0]] &, n + {0, 2}]], {n, 0, 78}] (* Michael De Vlieger, Jul 12 2017 *)

A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
A093101(n) = gcd(n!,A000522(n));
m1=m2=1; for(n=0,4096,m=m1; m1=m2; m2 = A093101(n+2); m124781 = gcd(m,m2); write("b093101.txt", n, " ", m); write("b124781.txt", n, " ", m124781); write("b123901.txt", n, " ", (n+3)/m124781)); \\ Antti Karttunen, Jul 12 2017

a(n) = (n+3)/A124781(n) = (n+3)/gcd(A093101(n),A093101(n+2)) where A093101(n) = gcd(n!,1+n+n(n-1)+...+n!).

A185108 a(0)=0; for n>0, a(n) = (n+2)*a(n-1) + 1.

Original entry on oeis.org

0, 1, 5, 26, 157, 1100, 8801, 79210, 792101, 8713112, 104557345, 1359245486, 19029436805, 285441552076, 4567064833217, 77640102164690, 1397521838964421, 26552914940324000, 531058298806480001, 11152224274936080022

Offset: 0

Olivier Gérard, Nov 02 2012

nonn

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

Cf. A000522, A002627, A056542.

[n le 1 select 0 else (n+1) * Self(n-1) + 1: n in [1..20]]; // Vincenzo Librandi, Dec 22 2012

RecurrenceTable[{a[0]==0, a[n]==(n+2)*a[n-1] + 1}, a, {n, 20}] (* Vincenzo Librandi, Dec 23 2012 *)
nxt[{n_,a_}]:={n+1,a(n+3)+1}; NestList[nxt,{0,0},20][[;;,2]] (* Harvey P. Dale, Aug 03 2023 *)

a(n) = e*Gamma(n+3,1)-(5/2)*(n+2)!, where Gamma(a,x) is the incomplete gamma function. [Bruno Berselli, Dec 24 2012]
Recurrence: a(n) = (n+3)*a(n-1) - (n+1)*a(n-2). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ (exp(1)-5/2)*sqrt(2*Pi)*exp(-n)*n^(n+5/2). - Vaclav Kotesovec, Aug 13 2013
From Peter Bala, Oct 09 2013: (Start)
a(n) = A000522(n+2) - 5/2*(n + 2)! = (n + 2)!*( (sum {k = 0..n + 2} 1/k!) - 5/2 ).
a(n) = floor((n + 2)!*(e - 5/2)).
E.g.f.: ((x^2 - 4*x + 5)*exp(x) - 5)/(1 - x)^3 = x + 5*x^2/2! + 26*x^3/3! + ....
1/(e - 5/2) = 3! - 3!/(1*5) - 4!/(5*26) - 5!/(26*157) - 6!/(157*1100) - .... (see A002627, A056542). (End)

Edited by Vincenzo Librandi and N. J. A. Sloane, Dec 24 2012

A248668 Sum of the numbers in the n-th row of the array at A248664.

Original entry on oeis.org

1, 4, 26, 260, 3610, 64472, 1409006, 36432076, 1087911890, 36844580000, 1395429571222, 58439837713556, 2681526361893626, 133783187672365480, 7210345924097089790, 417482356526745344732, 25844171201928905477026, 1703359919973405018460976

Offset: 1

Clark Kimberling, Oct 11 2014

The polynomial p(n,x) is defined as the numerator when the sum 1 + 1/(n*x + 1) + 1/((n*x + 1)(n*x + 2)) + ... + 1/((n*x + 1)(n*x + 2)...(n*x + n - 1)) is written as a fraction with denominator (n*x + 1)(n*x + 2)...(n*x + n - 1). For more, see A248664.

			The first six polynomials:
p(1,x) = 1
p(2,x) = 2 (x + 1)
p(3,x) = 9x^2 + 12 x +  5
p(4,x) = 4 (16 x^3 + 28 x^2 + 17 x + 4)
p(5,x) = 5 (125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
p(6,x) = 2 (3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that
a(1) = p(1,1) = 1, a(2) = p(2,1) = 4, a(3) = p(3,1) = 26.

Seiichi Manyama, Table of n, a(n) for n = 1..366

Cf. A000522, A248664, A248665, A248666, A248669, A294039.

with (combinat):
seq(add( k!*binomial(2*n-1,k),k = 0..n-1 ), n = 0..20);
# Peter Bala, Nov 14 2017

t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
c[n_] := c[n] = CoefficientList[p[x, n], x];
TableForm[Table[c[n], {n, 1, 10}]] (* A248664 array *)
Table[Apply[Plus, c[n]], {n, 1, 60}] (* A248668 *)

a(n) = sum(k = 0, n-1, k!*binomial(2*n-1,k)); \\ Michel Marcus, Nov 15 2017

a(n) = p(n,1), where p(n,x) is defined at A248664.
a(n) = Sum_{k = 0..n-1} k!*binomial(2*n-1,k). - Peter Bala, Nov 14 2017
a(n) = A294039(n) - Pochhammer(n, n)*A000522(n). - Peter Luschny, Nov 14 2017

A056546 a(n) = 5na(n-1) + 1 with a(0)=1.

Original entry on oeis.org

1, 6, 61, 916, 18321, 458026, 13740781, 480927336, 19237093441, 865669204846, 43283460242301, 2380590313326556, 142835418799593361, 9284302221973568466, 649901155538149792621, 48742586665361234446576

Offset: 0

Henry Bottomley, Jun 20 2000

			a(2) = 5*2*a(1) + 1 = 10*6 + 1 = 61.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000522, A010844, A010845, A056545, A056547 for analogs. A056546/(A000142*A000351) is an increasingly good approximation to 5th root of e.

m = 16; CoefficientList[E^x/(1-5x) + O[x]^m, x] Range[0, m-1]! (* Jean-François Alcover, Jun 03 2019 *)

a(n) = floor(e^(1/5)*5^n*n!).
From Philippe Deléham, Mar 14 2004: (Start)
a(n) = n!*Sum_{k=0..n} 5^(n-k)/k!.
E.g.f.: exp(x)/(1 - 5*x). (End)
a(n) = Sum_{k=0..n} P(n, k)*5^k. - Ross La Haye, Aug 29 2005
a(n) = hypergeometric_U(1, n+2 , 1/5)/5. - Peter Luschny, Nov 26 2014
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x >= 0} (5*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 5*x) satisfies the differential equation (1 - 5*x)*y' = (6 - 5*x)*y.
a(n) = (5*n + 1)*a(n-1) - 5*(n - 1)*a(n-2).
The sequence b(n) := 5^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 5. This leads to the continued fraction representation a(n) = 5^n*n!*( 1 + 1/(5 - 5/(11 - 10/(16 - ... - (5*n - 5)/(5*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/5) = 1 + 1/(5 - 5/(11 - 10/(16 - ... - (5*n - 5)/((5*n + 1) - ... )))). Cf. A010844. (End)

More terms from James Sellers, Jul 04 2000

cp's OEIS Frontend

A038154 a(n) = n! * Sum_{k=0..n-2} 1/k!.

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A046662 Sum of mistyped version of binomial coefficients.

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A054091 Row sums of A054090.

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Magma

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PARI

PARI

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A064383 Integers n >= 1 such that n divides 0!-1!+2!-3!+4!-...+(-1)^{n-1}(n-1)!.

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Extensions

A080954 E.g.f. exp(5x)/(1-x).

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Maple

Mathematica

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A096341 E.g.f.: exp(x)/(1-x)^7.

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Mathematica

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A123901 a(n) = (n+3)/gcd(d(n), d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0..n} 1/k! to lowest terms.

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A056546 a(n) = 5na(n-1) + 1 with a(0)=1.