A014288 -id:A014288 - 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 A published 2004 proof was retracted in 2011.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

A067078 a(1) = 1, a(2) = 2, a(n) = (n-1)a(n-1) - (n-2)a(n-2).

Original entry on oeis.org

1, 2, 3, 5, 11, 35, 155, 875, 5915, 46235, 409115, 4037915, 43954715, 522956315, 6749977115, 93928268315, 1401602636315, 22324392524315, 378011820620315, 6780385526348315, 128425485935180315, 2561327494111820315

Offset: 1

Amarnath Murthy, Jan 05 2002

nonn

Successive differences are factorials, or (n+1)st successive difference divided by n-th successive difference = n. I.e., {a(n+2)-a(n+1)}/{a(n+1)-a(n)} = n. - Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 14 2003

Equals the row sums of A165680. - Johannes W. Meijer, Oct 16 2009

			a(6) = 35, a(5)= 11 hence a(7) = 6*35 - 5*11 = 155.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100

Cf. A003422, A014288, A007489, A165680.

a067078 n = a067078_list !! (n-1)
a067078_list = scanl (+) 1 a000142_list
-- Reinhard Zumkeller, Dec 27 2011

a[1] = 1; a[2] = 2; a[n_] := a[n] = (n - 1)*a[n - 1] - (n - 2)*a[n - 2]; Table[ a[n], {n, 1, 25} ]
a=FoldList[Plus,2,(Range@40)! ];PrependTo[a,1] (* Vladimir Joseph Stephan Orlovsky, May 21 2010 *)

A067078(n)=sum(k=0, n-2, k!, 1) \\ M. F. Hasler, Dec 16 2007

a(n) = 1 + Sum_{i=0..n-2} i! = 2*A014288(n-1)+1 = A007489(n-2)+2 (n>1). - Henry Bottomley, Oct 23 2002; corrected by M. F. Hasler, Dec 16 2007
a(n) = 1+!(n-1) = 1+A003422(n-1); a(n+1)=a(n)+(n-1)!. - M. F. Hasler, Dec 16 2007
E.g.f.: A(x)=x*B(x) satisfies the differential equation B'(x)=B(x)+log(1/(1-x))+1. - Vladimir Kruchinin, Jan 19 2011

More terms from Robert G. Wilson v, Jan 07 2002

Edited by M. F. Hasler, Dec 16 2007

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

A100614 Numbers n such that (!n)/2 is prime, where !n = Sum_{k=0..n-1} k!.

Original entry on oeis.org

3, 4, 5, 8, 9, 10, 11, 30, 76, 163, 271, 273, 354, 721, 1796, 3733, 4769, 9316, 12221, 41532

Offset: 1

R. K. Guy, Dec 02 2004

No other terms below 50000. - Serge Batalov, Jul 23 2017

R. K. Guy, Unsolved Problems In Number Theory, B44.

Yves Gallot, Is the number of primes (of half the left handed factorials) finite?.
Eric Weisstein's World of Mathematics, Left Factorial
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A014288, Left factorials: A003422.

See A124375 for another version.

s = 1; Do[s = s + n!; If[ PrimeQ[s/2], Print[n + 1]], {n, 10^3}] (* Robert G. Wilson v, Dec 02 2004 *)

When A014288(n-1) is prime.

a(14) from Robert G. Wilson v, Dec 02 2004
a(15)=1796 from Ray Chandler, Dec 02 2004
a(17) from T. D. Noe, Dec 04 2004
Corrected by adding a(16)=3733 from Eric W. Weisstein, Oct 29 2005
a(18)=9316 from Eric W. Weisstein, Dec 27 2005
a(19)=12221 from Eric W. Weisstein, Oct 19 2006
a(20)=41532 from Serge Batalov, Jul 22 2017

A093468 a(1) = 1, a(2) = 2; for n >= 2, a(n+1) = a(n) + Sum {a(n)-a(i), i = 1 to n}.

Original entry on oeis.org

1, 2, 3, 6, 18, 78, 438, 2958, 23118, 204558, 2018958, 21977358, 261478158, 3374988558, 46964134158, 700801318158, 11162196262158, 189005910310158, 3390192763174158, 64212742967590158, 1280663747055910158

Offset: 1

Amarnath Murthy, Apr 07 2004

nonn

Ignoring the different offsets, this is the partial sums of A001710. - Jonathan Vos Post and Franklin T. Adams-Watters, May 12 2010

Cf. A093467.

a[1] = 1; a[2] = 2; a[n_] := a[n] = a[n - 1] + Sum[a[n - 1] - a[i], {i, n - 1}]; Table[ a[n], {n, 21}] (* Robert G. Wilson v, Apr 08 2004 *)

a(n) = A014288(n-1)+1. - Vladeta Jovovic, Apr 08 2004

a(n)=n*a(n-1)-(n-1)*a(n-2), n>3. - Gary Detlefs, Dec 30 2010

More terms from Robert G. Wilson v, Apr 08 2004

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

The number of all permutations of elements of sets {1..k}, k <= n, is b(n) = Sum_{k=0..n} k! while the number of all permutations of elements of all subsets of set {1,2..n} is c(n) = Sum_{k=0..n} binomial(n,k)!. So the required probability (in a sample space) is b(n)/c(n), n >= 1 (after reduction of the fractions).

Apparently a(n) = A014288(n) for n > 2. - Georg Fischer, Oct 23 2018

The sum of all terms in row m is (m+1)!/2. So a(m,n) = a(m-1,n) + m!/2, or is m!/2 if n=m.
Sum of m-th row = A001710(m+1). [Klaus Brockhaus, Jun 02 2009]
First column is A014288. - Franklin T. Adams-Watters, Oct 19 2013

cp's OEIS Frontend

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

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A067078 a(1) = 1, a(2) = 2, a(n) = (n-1)*a(n-1) - (n-2)*a(n-2).

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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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Magma

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PARI

PARI

PARI

Sage

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A100614 Numbers n such that (!n)/2 is prime, where !n = Sum_{k=0..n-1} k!.

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A093468 a(1) = 1, a(2) = 2; for n >= 2, a(n+1) = a(n) + Sum {a(n)-a(i), i = 1 to n}.

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Mathematica

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A117397 Column 3 of triangle A117396.

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A067078 a(1) = 1, a(2) = 2, a(n) = (n-1)a(n-1) - (n-2)a(n-2).

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.