A056199 -id:A056199 - cp's OEIS frontend

A007489 a(n) = Sum_{k=1..n} k!.

0, 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913, 43954713, 522956313, 6749977113, 93928268313, 1401602636313, 22324392524313, 378011820620313, 6780385526348313, 128425485935180313, 2561327494111820313, 53652269665821260313, 1177652997443428940313

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Equals row sums of triangle A143122 starting (1, 3, 9, 33, ...). - Gary W. Adamson, Jul 26 2008
a(n) for n>=4 is never a perfect square. - Alexander R. Povolotsky, Oct 16 2008
Number of cycles that can be written in the form (j,j+1,j+2,...), in all permutations of {1,2,...,n}. Example: a(3)=9 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), (132) we have 3+2+2+1+1+0=9 such cycles. - Emeric Deutsch, Jul 14 2009
Conjectured to be the length of the shortest word over {1,...,n} that contains each of the n! permutations as a factor (cf. A180632) [see Johnston]. - N. J. A. Sloane, May 25 2013
The above conjecture has been disproven for n>=6. See A180632 and the Houston 2014 reference. - Dmitry Kamenetsky, Mar 07 2016
a(n) is also the number of compositions of n if cardinal values do not matter but ordinal rankings do. Since cardinal values do not matter, a sequence of k summands summing to n can be represented as (s(1),...,s(k)), where the s's are positive integers and the numbers in parentheses are the initial ordinal rankings. The number of compositions of these summands are equal to k!, with k ranging from 1 to n. - Gregory L. Simay, Jul 31 2016
When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the left. Compare array A211370 for circular shifts to the left in a broader sense. Compare sequence A001563 for circular shifts to the right. - Tilman Piesk, Apr 29 2017
Since a(n) = (1!+2!+3!+...+n!) = 3(1+3!/3+4!/3+...+n!/3) is a multiple of 3 for n>2, the only prime in this sequence is a(2) = 3. - Eric W. Weisstein, Jul 15 2017
Generalization of 2nd comment: a(n) for n>=4 is never a perfect power (A007916) (Chentzov link). - Bernard Schott, Jan 26 2023

			a(4) = 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33. - _Michael B. Porter_, Aug 03 2016

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Section B44, Springer 2010.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Carauleanu Marc, Table of n, a(n) for n = 0..212 (first 100 terms from T. D. Noe)
N. N. Chentzov, D. O. Shklarsky, and I. M. Yaglom, The USSR Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics, problem 115, pp. 28 and 201-202, Dover publications, Inc., New York, 1993.
R. K. Guy, Letter to N. J. A. Sloane, 1987
Robin Houston, Tackling the Minimal Superpermutation Problem, arXiv:1408.5108 [math.CO], 2014.
Nathaniel Johnston, The minimal superpermutation problem (2013)
Nathaniel Johnston, Non-uniqueness of minimal superpermutations, Discrete Math. 313 (2013), no. 14, 1553--1557. MR3047396
S. Legendre and P. Paclet, On the Permutations Generated by Cyclic Shift , J. Int. Seq. 14 (2011) # 11.3.2.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
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.
Eric Weisstein's World of Mathematics, Factorial
Eric Weisstein's World of Mathematics, Left Factorial
G. Xiao, Sigma Server, Operate on "n!"
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 5.
Index entries for sequences related to factorial numbers

Equals A003422(n+1) - 1.
Cf. A000142, A000670, A001597, A007916, A137593, A143122, A161128, A180632.
Column k=0 of A120695.

List([1..20],n->Sum([1..n],Factorial)); # Muniru A Asiru, Jan 31 2018

a007489 n = a007489_list !! n
a007489_list = scanl (+) 0 $ tail a000142_list
-- Reinhard Zumkeller, Aug 29 2014

[0] cat [&+[Factorial(i): i in [1..n]]: n in [1..25]]; // Vincenzo Librandi, Sep 02 2016

A007489 := proc(n) local i; add(i!,i=1..n); end proc;

FoldList[Plus, 0, (Range@ 21)! ] (* Robert G. Wilson v, Sep 21 2007 *)
Table[Sum[i!, {i, 1, n}], {n, 0, 21}] (* Zerinvary Lajos, Jul 12 2009 *)
Accumulate[Range[50]!]  (* Harvey P. Dale, Apr 30 2011 *)
Table[Plus@@(Range[n]!), {n, 20}] (* Alonso del Arte, Jul 18 2011 *)

a(n)=sum(k=1,n,k!) \\ Charles R Greathouse IV, Jul 25 2011

a(n) = Sum_{k=1..n} P(n, k)/C(n, k). - Ross La Haye, Sep 21 2004
a(n) = 3*A056199(n) for n>=2. - Philippe Deléham, Feb 10 2007
a(n) = !(n+1)-1=A003422(n+1)-1. - Artur Jasinski, Nov 08 2007 [corrected by Werner Schulte, Oct 20 2021]
Starting (1, 3, 9, 33, 153, ...), = row sums of triangle A137593 - Gary W. Adamson, Jan 28 2008
a(n) = a(n-1) + n! for n >= 1. - Jaroslav Krizek, Jun 16 2009
E.g.f. A(x) satisfies to the differential equation A'(x)=A(x)+x/(1-x)^2+1. - Vladimir Kruchinin, Jan 22 2011
a(0)=0, a(1)=1, a(n) = (n+1)*a(n-1)-n*a(n-2). - Sergei N. Gladkovskii, Jul 05 2012
G.f.: W(0)*x/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
G.f.: x /(1-x)/Q(0),m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
E.g.f.: exp(x-1)*(Ei(1) - Ei(1-x)) - exp(x) + 1/(1 - x), where Ei(x) is the exponential integral. - Ilya Gutkovskiy, Nov 27 2016
a(n) = sqrt(a(n-1)*a(n+1)-a(n-2)*n*n!), n >= 2. - Gary Detlefs, Oct 26 2020
a(n) ~ n!. - Ridouane Oudra, Jun 11 2025

A117396 Triangle, read by rows, defined by: T(n,k) = (k+1)T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)T(n,j+k+1) for n>k with T(n,n)=1 for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 17, 11, 4, 1, 1, 77, 51, 19, 5, 1, 1, 437, 291, 109, 29, 6, 1, 1, 2957, 1971, 739, 197, 41, 7, 1, 1, 23117, 15411, 5779, 1541, 321, 55, 8, 1, 1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1, 1, 2018957, 1345971, 504739, 134597, 28041, 4807, 701, 89, 10, 1

Offset: 0

Paul D. Hanna, Mar 11 2006

Columns equal the partial sums of columns of triangle A092582 for k>0: T(n, k) - T(n-1, k) = A092582(n,k) = number of permutations p of [n] having length of first run equal to k.

			Triangle begins:
  1;
  1,      1;
  1,      2,      1;
  1,      5,      3,     1;
  1,     17,     11,     4,     1;
  1,     77,     51,    19,     5,    1;
  1,    437,    291,   109,    29,    6,   1;
  1,   2957,   1971,   739,   197,   41,   7,  1;
  1,  23117,  15411,  5779,  1541,  321,  55,  8, 1;
  1, 204557, 136371, 51139, 13637, 2841, 487, 71, 9, 1; ...
Matrix inverse is:
   1;
  -1,  1;
   1, -2,  1;
   1,  1, -3,  1;
   1,  1,  1, -4,  1;
   1,  1,  1,  1, -5, 1; ...
Matrix log is the integer triangle A117398:
    0;
    1,  0;
    0,  2,  0;
   -1,  2,  3,  0;
   -3,  4,  5,  4,  0;
   -9, 14, 15,  9,  5,  0;
  -33, 68, 65, 34, 14,  6,  0; ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A014288 (column 1), A056199 (column 2), A117397 (column 3), A003422 (row sums), A117398 (matrix log); A092582.

[k eq 0 select 1 else k*(&+[Factorial(j)/Factorial(k+1): j in [k-1..n]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 24 2021

T[n_, k_]:= T[n, k]= If[k==0, 1, k*Sum[j!/(k+1)!, {j,k-1,n}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 24 2021 *)

PARI
```
T(n,k)=if(n
				
```

/* Definition by Matrix Inverse: */ T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,if(r==c+1,-c,1))));(M^-1)[n+1,k+1]

PARI
```
T(n,k)=if(nPaul D. Hanna, Jun 20 2006
```

def A117396(n,k): return 1 if (k==0) else k*sum(factorial(j)/factorial(k+1) for j in (k-1..n))
flatten([[A117396(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 24 2021

T(n,k) = k*Sum_{j=k-1..n} j!/(k+1)! for n >= k > 0, with T(n,0) = 1 for n >= 0. - Paul D. Hanna, Jun 20 2006

A117397 Column 3 of triangle A117396.

Original entry on oeis.org

1, 4, 19, 109, 739, 5779, 51139, 504739, 5494339, 65369539, 843747139, 11741033539, 175200329539, 2790549065539, 47251477577539, 847548190793539, 16053185741897539, 320165936763977539, 6706533708227657539, 147206624680428617539, 3378708717041050697539

Offset: 0

Paul D. Hanna, Mar 11 2006

nonn

Equals the partial sums of column 3 of triangle A092582.

			G.f.: A(x) = 1 + 4*x + 19*x^2 + 109*x^3 + 739*x^4 + 5779*x^5 + 51139*x^6 + 504739*x^7 + 5494339*x^8 + 65369539*x^9 + 843747139*x^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A117396 (triangle), A014288 (column 1), A056199 (column 2), A003422 (row sums).

Cf. A003422, A007489, A092582.

[(&+[Factorial(j): j in [2..n+3]])/8: n in [0..30]]; // G. C. Greubel, Sep 05 2022

a:=n->sum(j!/8,j=2..n): seq(a(n), n=3..21); # Zerinvary Lajos, Jan 08 2007

Table[Sum[i!/8, {i, 2, n}], {n, 3, 21}] (* Zerinvary Lajos, Jul 12 2009 *)

{a(n)=1+sum(k=4,n+3,k!)*3/4!}
for(n=0,25,print1(a(n),", "))

[sum(factorial(j) for j in (2..n+3))/8 for n in (0..30)] # G. C. Greubel, Sep 05 2022

G.f. satisfies A(x) = (1-x)/(1 - 5*x + 5*x^2) * (1 + x^2*A'(x)).
a(n) = 1 + Sum_{k=4..n+3} k!*3/4! for n > 0, with a(0)=1.
G.f.: W(0)/(8*x*(1-x)) -1/(4*x), where W(k) = 1 + 1/( 1 - x*(k+3)/( x*(k+3) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2013
G.f.: (Sum_{n>=0} (n+2)!*x^n)/(8*x*(1-x)) - 1/(4*x). - Sergei N. Gladkovskii, Aug 20 2013
a(n) = (1/8)*(A007489(n+3) - 1) = (1/8)*(A003422(n+4) - 2). - G. C. Greubel, Sep 05 2022

A309579 Maximum principal ratio of a strongly connected digraph on n nodes.

Original entry on oeis.org

2, 6, 22, 102, 582, 3942, 30822, 272742, 2691942, 29303142, 348637542, 4499984742, 62618845542, 934401757542, 14882928349542, 252007880413542, 4520257017565542, 85616990623453542, 1707551662741213542, 35768179777214173542, 785101998295619293542, 18019779824218937053542

Offset: 3

Sinan G. Aksoy, Aug 08 2019

nonn

The principal ratio of a strongly connected digraph is the ratio of largest to smallest entries in the stationary distribution of a simple random walk on that digraph.

S. Aksoy, F. Chung, X. Peng, Extreme values of the stationary distribution of random walks on directed graphs, arXiv:1602.01162 [math.CO], 2016.
S. Aksoy, F. Chung, X. Peng, Extreme values of the stationary distribution of random walks on directed graphs, Advances in Applied Mathematics, 81:128--155, 2016.

Cf. A056199.

a(n) = (2/3) * (n-1)! * ( n/(n-1) + (1/(n-1)!) * sum(i=1, n-3, i!)); \\ Michel Marcus, Aug 11 2019

a(n) = (2/3) * (n-1)! * ( n/(n-1) + (1/(n-1)!) * Sum_{i=1..n-3} i! ).

a(n) = 2 * A056199(n-1). - Alois P. Heinz, Aug 11 2019

cp's OEIS Frontend

A007489 a(n) = Sum_{k=1..n} k!.

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A117396 Triangle, read by rows, defined by: T(n,k) = (k+1)*T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)*T(n,j+k+1) for n>k with T(n,n)=1 for n>=0.

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A117397 Column 3 of triangle A117396.

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Magma

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A309579 Maximum principal ratio of a strongly connected digraph on n nodes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A117396 Triangle, read by rows, defined by: T(n,k) = (k+1)T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)T(n,j+k+1) for n>k with T(n,n)=1 for n>=0.