A001710 -id:A001710 - cp's OEIS frontend

A001720 a(n) = n!/24.

1, 5, 30, 210, 1680, 15120, 151200, 1663200, 19958400, 259459200, 3632428800, 54486432000, 871782912000, 14820309504000, 266765571072000, 5068545850368000, 101370917007360000, 2128789257154560000, 46833363657400320000, 1077167364120207360000

Offset: 4

N. J. A. Sloane

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=5) ~ exp(-x)/x*(1 - 5/x + 30/x^2 - 210/x^3 + 1680/x^4 - 15120/x^5 + 151200/x^6 - 1663200/x^7 + ...) leads to this sequence. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 4..300
S. Cerdá, A simple scheme to find the solutions to Brocard-Ramanujan Diophantine Equation n! + 1 = m^2, arXiv:1504.06694 [math.NT], 2015.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 264.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
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.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A000142, A049353, A049459, A051338, A245334.
Cf. A001710, A001715, A007696, A094638.
Cf. A094587, A130534, A138533, A163931, A173333, A213936.

a001720 = (flip div 24) . a000142 -- Reinhard Zumkeller, Aug 31 2014

[Factorial(n)/24: n in [4..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_]:=n!/24; (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
Range[4,30]!/24 (* Harvey P. Dale, Jul 27 2021 *)

a(n)=n!/24 \\ Charles R Greathouse IV, Jan 12 2012

a(n)= A049353(n-3, 1) (first column of triangle).
E.g.f. if offset 0: 1/(1-x)^5.
a(n) = A173333(n,4). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n,n-4) / 5. - Reinhard Zumkeller, Aug 31 2014
G(x) = (1 - (1 + x)^(-4)) / 4 = x - 5 x^2/2! + 30 x^3/3! - ..., an e.g.f. for this signed sequence (for n!/4!), is the compositional inverse of H(x) = (1 - 4*x)^(-1/4) - 1 = x + 5 x^2/2! + 45 x^3/3! + ..., an e.g.f. for A007696. Cf. A094638, A001710 (for n!/2!), and A001715 (for n!/3!). Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
E.g.f.: x^4 / (4! * (1 - x)). - Ilya Gutkovskiy, Jul 09 2021
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=4} 1/a(n) = 24*e - 64.
Sum_{n>=4} (-1)^n/a(n) = 24/e - 8. (End)

A000153 a(n) = na(n-1) + (n-2)a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, 7, 32, 181, 1214, 9403, 82508, 808393, 8743994, 103459471, 1328953592, 18414450877, 273749755382, 4345634192131, 73362643649444, 1312349454922513, 24796092486996338, 493435697986613143, 10315043624498196944

Offset: 0

N. J. A. Sloane

With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=2 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, pp. 201-202. - Jaap Spies, Dec 12 2003
Starting (1, 2, 7, 32, ...) = inverse binomial transform of A001710 starting (1, 3, 12, 60, 360, 2520, ...). - Gary W. Adamson, Dec 25 2008
This sequence appears in Euler's analysis of the divergent series 1 - 1! + 2! - 3! + 4! ..., see Sandifer. For information about this and related divergent series see A163940. - Johannes W. Meijer, Oct 16 2009
a(n+1)=:b(n), n>=1, enumerates the ways to distribute n beads labeled differently from 1 to n, over a set of (unordered) necklaces, excluding necklaces with exactly one bead, and two indistinguishable, ordered, fixed cords, each allowed to have any number of beads. Beadless necklaces as well as beadless cords contribute each a factor 1 in the counting, e.g., b(0):= 1*1 = 1. See A000255 for the description of a fixed cord with beads.
This produces for b(n) the exponential (aka binomial) convolution of the subfactorial sequence {A000166(n)} and {(n+1)!}={A000042(n+1)}. This follows from the general problem with only k indistinguishable, ordered, fixed cords which has e.g.f. 1/(1-x)^k, and the pure necklace problem (no necklaces with one bead allowed) with e.g.f. for the subfactorials. Therefore also the recurrence b(n) = (n+1)*b(n-1) + (n-1)*b(n-2) with b(-1)=0 and b(0)=1 holds.
This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 02 2010
a(n) is a function of the subfactorials..sf... A000166(n) a(n) = (n*sf(n+1) - (n+1)*sf(n))/(2*n*(n-1)*(n+1)),n>1, with offset 1. - Gary Detlefs, Nov 06 2010
For even k the sequence a(n) (mod k) is purely periodic with exact period a divisor of k, while for odd k the sequence a(n) (mod k) is purely periodic with exact period a divisor of 2*k. See A047974. - Peter Bala, Dec 04 2017

			Necklaces and 2 cords problem. For n=4 one considers the following weak 2 part compositions of 4: (4,0), (3,1), (2,2), and (0,4), where (1,3) does not appear because there are no necklaces with 1 bead. These compositions contribute respectively sf(4)*1,binomial(4,3)*sf(3)*c2(1), (binomial(4,2)*sf(2))*c2(2), and 1*c2(4) with the subfactorials sf(n):=A000166(n) (see the necklace comment there) and the c2(n):=(n+1)! numbers for the pure 2 cord problem (see the above given remark on the e.g.f. for the k cords problem; here for k=2: 1/(1-x)^2). This adds up as 9 + 4*2*2 + (6*1)*6 + 120 = 181 = b(4) = A000153(5). - _Wolfdieter Lang_, Jun 02 2010
G.f. = x + 2*x^2 + 7*x^3 + 32*x^4 + 181*x^5 + 1214*x^6 + 9403*x^7 + 82508*x^8 + ...

Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 188.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
Anders Claesson, Giulio Cerbai, Dana C. Ernst, and Hannah Golab, Pattern-avoiding Cayley permutations via combinatorial species, arXiv:2407.19583 [math.CO], 2024.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 41.
Simon Plouffe, Exact formulas for integer sequences.
Ed Sandifer, Divergent Series, How Euler Did It, MAA Online, June 2006. - From _Johannes W. Meijer_, Oct 16 2009
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.

Cf. A000255, A000261, A001909, A001910, A090010, A055790, A090012-A090016.
Cf. A001710. - Gary W. Adamson, Dec 25 2008
a(n) = A086764(n + 1, 2). A000255 (necklaces with one cord). - Wolfdieter Lang, Jun 02 2010

a000153 n = a000153_list !! n
a000153_list = 0 : 1 : zipWith (+)
   (zipWith (*) [0..] a000153_list) (zipWith (*) [2..] $ tail a000153_list)
-- Reinhard Zumkeller, Mar 05 2012

f:= n-> floor(((n+1)!+1)/e): g:=n-> (n*f(n+1)-(n+1)*f(n))/(2*n*(n-1)*(n+1)):seq( g(n), n=2..20); # Gary Detlefs, Nov 06 2010
a := n -> `if`(n=0,0,hypergeom([3,-n+1],[],1))*(-1)^(n+1); seq(simplify(a(n)), n=0..20); # Peter Luschny, Sep 20 2014
0, seq(simplify(KummerU(-n + 1, -n - 1, -1)), n = 1..20); # Peter Luschny, May 10 2022

nn = 20; Prepend[Range[0, nn]!CoefficientList[Series[Exp[-x]/(1 - x)^3, {x, 0, nn}], x], 0]  (* Geoffrey Critzer, Oct 28 2012 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==n a[n-1]+(n-2)a[n-2]},a,{n,20}] (* Harvey P. Dale, May 08 2013 *)
a[ n_] := If[ n < 1, 0, (n - 1)! SeriesCoefficient[ Exp[ -x] / (1 - x)^3, {x, 0, n - 1}]]; (* Michael Somos, Jun 01 2013 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1, 3}, {}, x / (x + 1)] x / (x + 1), {x, 0, n}]; (* Michael Somos, Jun 01 2013 *)

x='x+O('x^66); concat([0],Vec(x*serlaplace(exp(-x)/(1-x)^3)))  \\ Joerg Arndt, May 08 2013

it = sloane.A000153.gen(0,1,2); [next(it) for i in range(21)] # Zerinvary Lajos, May 15 2009

E.g.f.: ( 1 - x )^(-3)*exp(-x), for offset 1.
a(n) = round(1/2*(n^2 + 3*n + 1)*n!/exp(1))/n , n>=1. - Simon Plouffe, Mar 1993
a(n) = (1/2) * A055790(n). - Gary Detlefs, Jul 12 2010
G.f.: hypergeom([1,3],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011
G.f.: (1+x)^2/(2*x*Q(0)) - 1/(2*x) - 1, where Q(k) = 1 - 2*k*x - x^2*(k + 1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 08 2013
G.f.: -1/G(0), where G(k) = 1 + 1/(1 - (1+x)/(1 + x*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
G.f.: x/Q(0), where Q(k) = 1 - 2*x*(k+1) - x^2*(k+1)*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2013
a(n) = hypergeom([3, -n+1], [], 1)*(-1)^(n+1) for n>=1. - Peter Luschny, Sep 20 2014
a(n) = KummerU(-n + 1, -n - 1, -1) for n >= 1. - Peter Luschny, May 10 2022
a(n) = (n^2 + 3*n + 1)*Gamma(n,-1)/(2*exp(1)) + (1 + n/2)*(-1)^n for n >= 1. - Martin Clever, Apr 06 2023

A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 120, 60, 20, 5, 1, 720, 360, 120, 30, 6, 1, 5040, 2520, 840, 210, 42, 7, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 3628800, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Reinhard Zumkeller, Feb 19 2010

From Wolfdieter Lang, Jun 27 2012: (Start)

T(n-1,k), k=1,...,n-1, gives the number of representative necklaces with n beads (C_N symmetry) of n+1-k distinct colors, say c[1],c[2],...,c[n-k+1], corresponding to the color signature determined by the partition k,1^(n-k) of n. The representative necklaces have k beads of color c[1]. E.g., n=4, k=2: partition 2,1,1, color signature (parts as exponents) c[1]c[1]c[2]c[3], 3=T(3,2) necklaces (write j for color c[j]): cyclic(1123), cyclic(1132) and cyclic(1213). See A212359 for the numbers for general partitions or color signatures. (End)

			Triangle starts:
n\k      1       2      3      4     5    6   7  8  9 10 ...
1        1
2        2       1
3        6       3      1
4       24      12      4      1
5      120      60     20      5     1
6      720     360    120     30     6    1
7     5040    2520    840    210    42    7   1
8    40320   20160   6720   1680   336   56   8  1
9   362880  181440  60480  15120  3024  504  72  9  1
10 3628800 1814400 604800 151200 30240 5040 720 90 10  1
... - _Wolfdieter Lang_, Jun 27 2012

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Index entries for sequences related to factorial numbers.

Row sums give A002627.
Central terms give A006963:
T(2*n-1,n) = A006963(n+1).
T(2*n,n) = A001813(n).
T(2*n,n+1) = A001761(n).
1 < k <= n: T(n,k) = T(n,k-1) / k.
1 <= k <= n: T(n+1,k) = A119741(n,n-k+1).
1 <= k <= n: T(n+1,k+1) = A162995(n,k).
T(n,1) = A000142(n).
T(n,2) = A001710(n) for n>1.
T(n,3) = A001715(n) for n>2.
T(n,4) = A001720(n) for n>3.
T(n,5) = A001725(n) for n>4.
T(n,6) = A001730(n) for n>5.
T(n,7) = A049388(n-7) for n>6.
T(n,8) = A049389(n-8) for n>7.
T(n,9) = A049398(n-9) for n>8.
T(n,10) = A051431(n) for n>9.
T(n,n-7) = A159083(n+1) for n>7.
T(n,n-6) = A053625(n+1) for n>6.
T(n,n-5) = A052787(n) for n>5.
T(n,n-4) = A052762(n) for n>4.
T(n,n-3) = A007531(n) for n>3.
T(n,n-2) = A002378(n-1) for n>2.
T(n,n-1) = A000027(n) for n>1.
T(n,n) = A000012(n).
Cf. A138533, A002627.

a173333 n k = a173333_tabl !! (n-1) !! (k-1)
a173333_row n = a173333_tabl !! (n-1)
a173333_tabl = map fst $ iterate f ([1], 2)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

Table[n!/k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

E.g.f.: (exp(x*y) - 1)/(x*(1 - y)). - Olivier Gérard, Jul 07 2011

T(n,k) = A094587(n,k), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

A001711 Generalized Stirling numbers.

Original entry on oeis.org

1, 7, 47, 342, 2754, 24552, 241128, 2592720, 30334320, 383970240, 5231113920, 76349105280, 1188825724800, 19675048780800, 344937224217600, 6386713749964800, 124548748102195200, 2551797512248320000, 54804198761303040000, 1231237843834521600000

Offset: 0

N. J. A. Sloane

nonn

The asymptotic expansion of the higher order exponential integral E(x,m=2,n=3) ~ exp(-x)/x^2*(1 - 7/x + 47/x^2 - 342/x^3 + 2754/x^4 - 24552/x^5 + 241128/x^6 - ...) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009
For n > 4, a(n) mod n = 0 for n composite, = n-3 for n prime. - Gary Detlefs, Jul 18 2011
From Petros Hadjicostas, Jun 11 2020: (Start)
For nonnegative integers n, m and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) using slightly different notation. They were further examined by Mitrinovic and Mitrinovic (1962).
These numbers are defined via the g.f. Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0.
As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, and R_n^m(a,b) = 0 for n < m. (Because an empty product is by definition 1, we may let R_0^0(a,b) = 1.)
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m). (Array A008275 is the same as array A048994 but with no zero row and no zero column.)
We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current sequence, a(n) = R_{n+1}^1(a=-3, b=-1) for n >= 0. (End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..440[Terms 0 to 100 computed by T. D. Noe; terms 101 to 440 computed by G. C. Greubel, Jan 15 2017]
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356. [The numbers R_n^m(a,b) are introduced.]
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49. [Annotated scanned copy]
J. Riordan, Letter of 04/11/74.

Related to n!*the k-th successive summation of the harmonic numbers: k=0..A000254, k=1..A001705, k=2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564.

a := n-> add(1/2*((n+3)!/(k+3)), k=0..n): seq(a(n), n=0..19); # Zerinvary Lajos, Jan 22 2008
a := n -> (n+1)!*hs2(n+1): hs2 := n-> add(hs(k), k=0..n): hs := n-> add(h(k), k=0..n): h := n-> add(1/k, k=1..n): seq(a(n), n=0..19); # Gary Detlefs, Jan 01 2011

f[k_] := k + 2; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 1, 16}]; (* Clark Kimberling, Dec 29 2011 *)
Table[(n + 3)!*Sum[1/(2*k + 4), {k, 1, n + 1}], {n,0,100}] (* G. C. Greubel, Jan 15 2017 *)

for(n=0, 19, print1((n+1)! * sum(k=0, n, binomial(k + 2, 2) / (n + 1 - k)),", ")) \\ Indranil Ghosh, Mar 13 2017

R(n,m,a,b) =  sum(k=0, n-m, (-1)^k*a^k*b^(n-m-k)*binomial(m+k,k)*stirling(n, m+k,1));
aa(n) = R(n+1,1,-3,-1);
for(n=0, 19, print1(aa(n), ",")) \\ Petros Hadjicostas, Jun 11 2020

E.g.f.: -log(1 - x)/(1 - x)^3 if offset 1. With offset 0: (d/dx)(-log(1 - x)/(1 - x)^3) = (1 - 3*log(1 - x))/(1 - x)^4.
a(n) = Sum_{k=0..n} ((-1)^(n+k)*(k+1)*3^k*Stirling1(n+1, k+1)). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
a(n) = n!*Sum_{k=0..n-1} ((-1)^k*binomial(-3,k)/(n-k)). - Milan Janjic, Dec 14 2008
a(n) = ( A000254(n+3) - 3*A001710(n+3) )/2. - Gary Detlefs, May 24 2010
a(n) = ((n+3)!/4) * (2*h(n+3) - 3), where h(n) = Sum_{k=1..n} (1/k) is the n-th harmonic number. - Gary Detlefs, Aug 15 2010
a(n) = n!*[2]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n. With offset 1. - Gary Detlefs, Jan 04 2011
a(n) = (n+3)! * Sum_{k=1..n+1} (1/(2*k+4)). - Gary Detlefs, Sep 14 2011
a(n) = (n+1)! * Sum_{k=0..n} (binomial(k+2,2)/(n+1-k)). - Gary Detlefs, Dec 01 2011
a(n) = A001705(n+2) - A182541(n+4). - Anton Zakharov, Jul 02 2016
a(n) ~ n^(n+7/2) * exp(-n) * sqrt(Pi/2) * log(n) * (1 + (gamma - 3/2)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 12 2016
Conjectural D-finite with recurrence: a(n) + (-2*n-5)*a(n-1) + (n+2)^2*a(n-2)=0. - R. J. Mathar, Feb 16 2020
From Petros Hadjicostas, Jun 11 2020: (Start)
Since a(n) = R_{n+1}^1(a=-3, b=-1), it follows from Mitrinovic (1961) and Mitrinovic and Mitrinovic (1962) that:
a(n) = [x] Product_{r=0}^n (x + 3 + r) = (Product_{r=0}^n (3 + r)) * Sum_{s=0}^n 1/(3 + s).
a(n) = (n + 2)!/2 + (n + 3)*a(n-1) for n >= 1. [This can be used to prove R. J. Mathar's recurrence above.] (End)

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

Maple programs corrected and edited by Johannes W. Meijer, Nov 28 2012

A000261 a(n) = na(n-1) + (n-3)a(n-2), with a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 3, 13, 71, 465, 3539, 30637, 296967, 3184129, 37401155, 477471021, 6581134823, 97388068753, 1539794649171, 25902759280525, 461904032857319, 8702813980639617, 172743930157869827, 3602826440828270029, 78768746000235327495, 1801366114380914335441

Offset: 1

N. J. A. Sloane

nonn

With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=3 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, pp. 201-202. - Jaap Spies, Dec 12 2003
a(n+2)=:b(n), n>=1, enumerates the ways to distribute n beads, labeled differently from 1 to n, over a set of (unordered) necklaces, excluding necklaces with exactly one bead, and three indistinguishable, ordered, fixed cords, each allowed to have any number of beads. Beadless necklaces as well as a beadless cords contribute each a factor 1 in the counting, e.g., b(0):= 1*1 =1. See A000255 for the description of a fixed cord with beads.
This produces for b(n) the exponential (aka binomial) convolution of the subfactorial sequence {A000166(n)} and the sequence {A001710(n+2)}. See the necklaces and cords problem comment in A000153. Therefore also the recurrence b(n) = (n+2)*b(n-1) + (n-1)*b(n-2) with b(-1)=0 and b(0)=1 holds. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 02 2010

			Necklaces and 3 cords problem. For n=4 one considers the following weak 2 part compositions of 4: (4,0), (3,1), (2,2), and (0,4), where (1,3) does not appear because there are no necklaces with 1 bead. These compositions contribute respectively sf(4)*1,binomial(4,3)*sf(3)*c3(1), (binomial(4,2)*sf(2))*c3(2), and 1*c3(4) with the subfactorials sf(n):=A000166(n) (see the necklace comment there) and the c3(n):=A001710(n+2) = (n+2)!/2! numbers for the pure 3 cord problem (see the remark on the e.g.f. for the k cords problem in A000153; here for k=3: 1/(1-x)^3). This adds up as 9 + 4*2*3 + (6*1)*12 + 360 = 465 = b(4) = A000261(6). - _Wolfdieter Lang_, Jun 02 2010
G.f. = x^2 + 3*x^3 + 13*x^4 + 71*x^5 + 465*x^6 + 3539*x^7 + 30637*x^8 + ...

Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 188.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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=1..102
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), pp. 197-210.

Cf. A000255, A000153, A001909, A001910, A090010, A055790, A090012-A090016.
Cf. A086764(n+1,3), n>=1.
Cf. A000153 (necklaces and two cords). - Wolfdieter Lang, Jun 02 2010

a:= proc(n) a(n):= `if`(n<3, n-1, n*a(n-1) +(n-3)*a(n-2)) end:
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 03 2012
a := n -> `if`(n=1,0,hypergeom([4,-n+2],[],1))*(-1)^(n); seq(round(evalf(a(n), 100)), n=1..22); # Peter Luschny, Sep 20 2014

nn=20;Prepend[Range[0,nn]!CoefficientList[Series[Exp[-x]/ (1-x)^4, {x,0,nn}],x],0]  (* Geoffrey Critzer, Nov 03 2012 *)
a[ n_] := SeriesCoefficient[ x^2 HypergeometricPFQ[ {1, 4}, {}, x / (1 + x)] / (1 + x), {x, 0, n}]; (* Michael Somos, May 04 2014 *)
a[ n_] := If[ n < 2, 0, With[{m = n - 1}, Round[ Gamma[m] (m^3 + 6 m^2 + 8 m + 1) Exp[-1]/6]]]; (* Michael Somos, May 04 2014 *)

E.g.f.: exp(-x)/(1-x)^4 for offset -1.
For offset -1: (1/6)*Sum_{k=0..n} (-1)^k*(n-k+1)*(n-k+2)*(n-k+3)*n!/k! = (1/6)*(A000166(n)+3*A000166(n+1)+3*A000166(n+2)+A000166(n+3)). - Vladeta Jovovic, Jan 07 2003
a(n+1) = round( GAMMA(n)*(n^3+6*n^2+8*n+1)*exp(-1)/6 ) for n>0. - Mark van Hoeij, Nov 11 2009
G.f.: x^2*hypergeom([1,4],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011
E.g.f. for offset -1: 1/(exp(x)*(1-x)^4) = 1/E(0) where E(k) = 1 - 4*x/(1 + 3*x/(2 - 3*x + 4*x/(3 - 2*x + 3*x/(4 - x - 4/(1 + x^3*(k+1)/E(k+1)))))); (continued fraction, 3rd kind, 6-step). - Sergei N. Gladkovskii, Sep 21 2012
a(n) = hypergeometric([4,-n+2],[],1)*(-1)^n for n>=2. - Peter Luschny, Sep 20 2014

More terms from Vladeta Jovovic, Jan 07 2003

A049388 a(n) = (n+7)!/7!.

Original entry on oeis.org

1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200, 4151347200, 70572902400, 1270312243200, 24135932620800, 482718652416000, 10137091700736000, 223016017416192000, 5129368400572416000, 123104841613737984000, 3077621040343449600000, 80018147048929689600000

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher order exponential integral E(x,m=1,n=8) ~ exp(-x)/x*(1 - 8/x + 72/x^2 - 720/x^3 + 7920/x^4 - 95040/x^5 + 235520/x^6 - 17297280/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A051339, A051379.

Cf. A130534, A163931, A173333, A245334.

a049388 = (flip div 5040) . a000142 . (+ 7)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+7)/5040: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

((Range[0,20]+7)!)/7! (* Harvey P. Dale, Jul 31 2012 *)

vector(20,n,n--; (n+7)!/7!) \\ G. C. Greubel, Aug 15 2018

a(n)= A051379(n, 0)*(-1)^n (first unsigned column of triangle).
a(n) = (n+7)!/7!.
E.g.f.: 1/(1-x)^8.
a(n) = A173333(n+7,7). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+7,n) / 8. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 5040*e - 13699.
Sum_{n>=0} (-1)^n/a(n) = 1855 - 5040/e. (End)

A055314 Triangle T(n,k) read by rows: number of labeled trees with n nodes and k leaves, n >= 2, 2 <= k <= n.

Original entry on oeis.org

1, 3, 0, 12, 4, 0, 60, 60, 5, 0, 360, 720, 210, 6, 0, 2520, 8400, 5250, 630, 7, 0, 20160, 100800, 109200, 30240, 1736, 8, 0, 181440, 1270080, 2116800, 1058400, 151704, 4536, 9, 0, 1814400, 16934400, 40219200, 31752000, 8573040, 695520, 11430, 10, 0

Offset: 2

Christian G. Bower, May 11 2000

			Triangle T(n,k) begins:
     1;
     3,    0;
    12,    4,    0;
    60,   60,    5,   0;
   360,  720,  210,   6, 0;
  2520, 8400, 5250, 630, 7, 0;
  ...

Moon, J. W. Various proofs of Cayley's formula for counting trees. 1967 A seminar on Graph Theory pp. 70--78 Holt, Rinehart and Winston, New York; MR0214515 (35 #5365). - From N. J. A. Sloane, Jun 07 2012
Renyi, Alfred. Some remarks on the theory of trees. Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 1959 73--85. MR0115938 (22 #6735). - From N. J. A. Sloane, Jun 07 2012
D. Stanton and D. White, Constructive Combinatorics, Springer, 1986; see p. 67.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
A. M. Hamel, Priority queue sorting and labeled trees, Annals Combin., 7 (2003), 49-54.
F. Harary, A. Mowshowitz and J. Riordan, Labeled trees with unlabeled end-points, J. Combin. Theory, 6 (1969), 60-64. - From _N. J. A. Sloane_, Jun 07 2012
Vites Longani, A formula for the number of labelled trees, Comp. Math. Appl. 56 (2008) 2786-2788.
John Riordan, The enumeration of labeled trees by degrees, Bull. Amer. Math. Soc. 72 1966 110--112. MR0186583 (32 #4042). - From _N. J. A. Sloane_, Jun 07 2012
Index entries for sequences related to trees

Cf. A000272.

Row sums give A000272. Columns 2 through 12: A001710, A055315-A055324.

T := (n,k) -> binomial(n+1,k)*add( binomial(n+1-k,i)*(-1)^(n+1-k-i)*i^(n-1), i=0..n+1-k);
# The following version gives the triangle for any n>=1, k>=1, based on the Harary et al. (1969) paper - N. J. A. Sloane, Jun 07 2012
with(combinat);
R:=proc(n,k)
if n=1 then if k=1 then RETURN(1) else RETURN(0); fi
    elif (n=2 and k=2) then RETURN(1)
    elif (n=2 and k>2) then RETURN(0)
    else stirling2(n-2,n-k)*n!/k!;
    fi;
end;

Table[Table[Binomial[n,k] Sum[(-1)^j Binomial[n-k,j] (n-k-j)^(n-2),{j,0,n-k}],{k,2,n-1}],{n,2,10}]//Grid (* Geoffrey Critzer, Nov 12 2011 *)
Table[(n!/k!)*StirlingS2[n - 2, n - k], {n, 2, 7}, {k, 2, n}]//Flatten (* G. C. Greubel, May 17 2017 *)

A055314(n,k) := block(
        n!/k!*stirling2(n-2,n-k)
)$
for n : 2 thru 10 do
        for k : 2 thru n do
                print(A055314(n,k)," ") ; /* R. J. Mathar, Mar 06 2012 */

for(n=2,20, for(k=2,n, print1((n!/k!)*stirling(n-2, n-k, 2), ", "))) \\ G. C. Greubel, May 17 2017

E.g.f.: A(x, y)=(1-x+x*y)*B(x, y)-B(x, y)^2/2. B(x, y): e.g.f. of A055302.
T(n, k) = binomial(n+1, k)*Sum( binomial(n+1-k, i)*(-1)^(n+1-k-i)*i^(n-1), i=0..n+1-k).
T(n, k) = (n!/k!)*Stirling2(n-2, n-k). - Vladeta Jovovic, Jan 28 2004

A145882 Triangle read by rows: T(n,k) is the number of even permutations of {1,2,...,n} having k descents (n >= 1, k >= 0).

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 5, 1, 1, 14, 30, 14, 1, 1, 29, 147, 155, 28, 1, 64, 586, 1208, 605, 56, 1, 127, 2133, 7819, 7819, 2133, 127, 1, 1, 262, 7288, 44074, 78190, 44074, 7288, 262, 1, 1, 517, 23893, 227569, 655315, 655039, 227623, 23947, 496, 1, 1044, 76332, 1101420

Offset: 1

Emeric Deutsch, Nov 11 2008

Number of entries in row n is 1+floor(binomial(n,2)/2)-floor(binomial(n-2,2)/2).

Sum of entries in row n is A001710(n) for n>=2.

			T(4,2) = 5 because we have 4132, 2143, 4213, 2431 and 3241.
Triangle begins with T(1,0):
  1
  1
  1    2
  1    5      5       1
  1   14     30      14       1
  1   29    147     155      28
  1   64    586    1208     605      56
  1  127   2133    7819    7819    2133     127       1
  1  262   7288   44074   78190   44074    7288     262     1
  1  517  23893  227569  655315  655039  227623   23947   496
  1 1044  76332 1101420 4869558 7862124 4868556 1102068 76305 992

Alois P. Heinz, Rows n = 1..142, flattened
J. Shareshian and M. L. Wachs, q-Eulerian polynomials: excedance number and major index, Electronic Research Announcements of the Amer. Math. Soc., 13 (2007), 33-45.
R. P. Stanley, Binomial posets, Möbius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336-356.
S. Tanimoto, A study of Eulerian numbers for permutations in the alternating group, Integers, Electronic J. of Combinatorial Number Theory, 6 (2006), #A31.

Cf. A001710, A145883.

Cf. A128612 (similar with rows reversed).

for n to 11 do qbr := proc (m) options operator, arrow; sum(q^i, i = 0 .. m-1) end proc; qfac := proc (m) options operator, arrow; product(qbr(j), j = 1 .. m) end proc; Exp := proc (z) options operator, arrow; sum(q^binomial(m, 2)*z^m/qfac(m), m = 0 .. 19) end proc; g := (1-t)/(Exp(z*(t-1))-t); gser := simplify(series(g, z = 0, 17)); a[n] := simplify(qfac(n)*coeff(gser, z, n)); b[n] := (a[n]+subs(q = -q, a[n]))*1/2; P[n] := sort(subs(q = 1, b[n])) end do; for n to 11 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*binomial(n, 2)) -floor((1/2)*binomial(n-2, 2))) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, t, expand(
       add(b(u+j-1, o-j, irem(t+j-1+u, 2)), j=1..o)+
       add(b(u-j, o+j-1, irem(t+u-j, 2))*x, j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))
        (add(b(j-1, n-j, irem(j, 2)), j=1..n)):
seq(T(n), n=1..12);  # Alois P. Heinz, Nov 19 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, t, Expand[Sum[b[u+j-1, o-j, Mod[t+j-1+u, 2]], {j, 1, o}] + Sum[b[u-j, o+j-1, Mod[t+u-j, 2]]*x, {j, 1, u}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]] [Sum[b[j-1, n-j, Mod[j, 2]], {j, 1, n}]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)
Needs["Combinatorica`"];
Table[(Eulerian[n, k] + Sum[Binomial[j-1-Floor[n/2], j] Eulerian[Ceiling[n/2], k-j], {j, Max[0, k-Ceiling[n/2]], Min[Floor[n/2], k]}])/2, {n, 25}, {k, 0, n-1}] // Flatten // DeleteCases[0] (* Robert A. Russell, Nov 14 2018 *)

In the Shareshian and Wachs reference (p. 35) a q-analog of the exponential g.f. of the Eulerian polynomials is given for the joint distribution of (inv, des) (see also the Stanley reference). The first Maple program given below makes use of this function by considering its even part.

T(n,k) = (euler(n,k) + Sum_{j=max(0, k+1-ceiling(n/2))..min(floor(n/2), k)} binomial(j-1-floor(n/2), j) * euler(ceiling(n/2), k-j)) / 2, where euler(n,k) is the Eulerian number A173018 (not A008292, which has different indexing). - Robert A. Russell, Nov 15 2018

A002135 Number of terms in a symmetrical determinant: a(n) = na(n-1) - (n-1)(n-2)*a(n-3)/2.

Original entry on oeis.org

1, 1, 2, 5, 17, 73, 388, 2461, 18155, 152531, 1436714, 14986879, 171453343, 2134070335, 28708008128, 415017867707, 6416208498137, 105630583492969, 1844908072865290, 34071573484225549, 663368639907213281, 13580208904207073801

Offset: 0

N. J. A. Sloane

a(n) is the number of collections of necklaces created by using exactly n different colored beads (to make the entire collection). - Geoffrey Critzer, Apr 19 2009
a(n) is the number of ways that a deck with 2 cards of each of n types may be dealt into n hands of 2 cards each, assuming that the order of the hands and the order of the cards in each hand are irrelevant. See the Art of Problem Solving link for proof. - Joel B. Lewis, Sep 30 2012
From Bruce Westbury, Jan 22 2013: (Start)
It follows from the respective exponential generating functions that A002135 is the binomial transform of A002137:
A002135(n) = Sum_{k=0..n} binomial(n,k)*A002137(k),
2 = 1.1 + 2.0 + 1.1,
5 = 1.1 + 3.0 + 3.1 + 1.1,
17 = 1.1 + 4.0 + 6.1 + 4.1 + 1.6, ...
A002137 arises from looking at the dimension of the space of invariant tensors of the r-th tensor power of the adjoint representation of the symplectic group Sp(2n) (for n large compared to r).
(End)
a(n) is the number of representations required for the symbolic central moments of order 2 for the multivariate normal distribution, that is,   E[X1^2 X2^2 .. Xn^2|mu=0, Sigma] (Phillips 2010). These representations are the upper-triangular, positive integer matrices for which for each i, the sum of the i-th row and i-th column equals 2, the power of each component. This can be shown starting from the formulation by Joel B Lewis. See "Proof for multivariate normal moments" link below for a proof. - Kem Phillips, Aug 20 2014
Equivalent to Critzer's comment, a(n) is the number of ways to cover n labeled vertices by disjoint undirected cycles, hence the exponential transform of A001710(n - 1). - Gus Wiseman, Oct 20 2018

			For n = 3, the a(3) = 5 ways to deal out the deck {1, 1, 2, 2, 3, 3} into three two-card hands are {11, 22, 33}, {12, 12, 33}, {13, 13, 22}, {11, 23, 23}, {12, 13, 23}. - _Joel B. Lewis_, Sep 30 2012

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260, #12, a_n.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.9 and Problem 5.22.

T. D. Noe, Table of n, a(n) for n=0..100
A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.
A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5. [Annotated scanned copy]
Art of Problem Solving, Partitioning a deck with 2 cards in n types into pairs
A. Cayley, On the number of distinct terms in a symmetrical or partially symmetrical determinant, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, pp. 185-190. [Annotated scanned copy]
Tomislav Došlic and Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - _N. J. A. Sloane_, May 01 2012
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41.
Toufik Mansour, The length of the initial longest increasing sequence in a permutation, Art Disc. Appl. Math. (2023).
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages]
K. Phillips, R functions to symbolically compute the central moments of the multivariate normal distribution, Journal of Statistical Software, Feb 2010.
Kem Phillips, Proof for multivariate normal moments
Richard Stanley and J. Riordan, Problem E2297, Amer. Math. Monthly, 79 (1972), 519-520.

A diagonal of A260338.
Row sums of A215771.
Column k=2 of A257463 and A333467.
Cf. A002137, A059422, A059423, A059424.
Cf. A001147, A007717, A037143, A001710, A215771, A319225, A319226, A320655, A320656.

G:=proc(n) option remember; if n <= 1 then 1 elif n=2 then
2 else n*G(n-1)-binomial(n-1,2)*G(n-3); fi; end;

a[x_]:=Log[1/(1-x)^(1/2)]+x/2+x^2/4;Range[0, 20]! CoefficientList[Series[Exp[a[x]], {x, 0, 20}], x]
RecurrenceTable[{a[0]==a[1]==1,a[2]==2,a[n]==n*a[n-1]-(n-1)(n-2)* a[n-3]/2}, a,{n,30}] (* Harvey P. Dale, Dec 16 2011 *)
Table[Sum[Binomial[k, i] Binomial[i - 1/2, n - k] (3^(k - i) n!)/(4^k k!) (-1)^(n - k - i), {k, 0, n}, {i, 0, k}], {n, 0, 12}] (* Emanuele Munarini, Aug 25 2017 *)

a(n):=sum(sum(binomial(k,i)*binomial(i-1/2,n-k)*(3^(k-i)*n!)/(4^k*k!)*(-1)^(n-k-i),i,0,k),k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 25 2017 */

a(n) = if(n<3, [1,1,2][n+1], n*a(n-1) - (n-1)*(n-2)*a(n-3)/2 ); /* Joerg Arndt, Apr 07 2013 */

E.g.f.: (1-x)^(-1/2)*exp(x/2+x^2/4).
D-finite with recurrence a(n+1) = (n+1)*a(n) - binomial(n, 2)*a(n-2). See Comtet.
Asymptotics: a(n) ~ sqrt(2)*exp(3/4-n)*n^n*(1+O(1/n)). - Pietro Majer, Oct 27 2009
E.g.f.: G(0)/sqrt(1-x) where G(k) = 1 + x*(x+2)/(4*(2*k+1) - 4*x*(x+2)*(2*k+1)/(x*(x+2) + 8*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 31 2013
a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(k,i)*binomial(i-1/2,n-k)*(3^(k-i)*n!)/(4^k*k!)*(-1)^(n-k-i). - Emanuele Munarini, Aug 25 2017

A049389 a(n) = (n+8)!/8!.

Original entry on oeis.org

1, 9, 90, 990, 11880, 154440, 2162160, 32432400, 518918400, 8821612800, 158789030400, 3016991577600, 60339831552000, 1267136462592000, 27877002177024000, 641171050071552000, 15388105201717248000, 384702630042931200000, 10002268381116211200000

Offset: 0

Wolfdieter Lang

The asymptotic expansion of the higher-order exponential integral E(x,m=1,n=9) ~ exp(-x)/x*(1 - 9/x + 90/x^2 - 990/x^3 + 11880/x^4 - 154440/x^5 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000142, A001710, A001715, A001720, A001725, A001730, A049388, A051379, A051380.

Cf. A130534, A163931, A173333, A245334.

a049389 = (flip div 40320) . a000142 . (+ 8)
-- Reinhard Zumkeller, Aug 31 2014

[Factorial(n+8)/40320: n in [0..25]]; // Vincenzo Librandi, Jul 20 2011

a[n_] := (n + 8)!/8!; Array[a, 20, 0] (* Amiram Eldar, Jan 15 2023 *)

PARI
```
a(n) = (n+8)!/8!;
```

a(n)= A051380(n, 0)*(-1)^n (first unsigned column of triangle).
a(n) = (n+8)!/8!.
E.g.f.: 1/(1-x)^9.
a(n) = A173333(n+8,8). - Reinhard Zumkeller, Feb 19 2010
a(n) = A245334(n+8,n) / 9. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=0} 1/a(n) = 40320*e - 109600.
Sum_{n>=0} (-1)^n/a(n) = 40320/e - 14832. (End)

cp's OEIS Frontend

A001720 a(n) = n!/24.

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A000153 a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.

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A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

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Haskell

Mathematica

Formula

A001711 Generalized Stirling numbers.

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Maple

Mathematica

PARI

PARI

Formula

Extensions

A000261 a(n) = n*a(n-1) + (n-3)*a(n-2), with a(1) = 0, a(2) = 1.

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Maple

Mathematica

Formula

Extensions

A049388 a(n) = (n+7)!/7!.

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A000153 a(n) = na(n-1) + (n-2)a(n-2), with a(0) = 0, a(1) = 1.

A000261 a(n) = na(n-1) + (n-3)a(n-2), with a(1) = 0, a(2) = 1.

A002135 Number of terms in a symmetrical determinant: a(n) = na(n-1) - (n-1)(n-2)*a(n-3)/2.