A265609 -id:A265609 - cp's OEIS frontend

A265890 Array read by ascending antidiagonals: A(n,k) = A099563(A265609(n,k)), with n as row >= 0, k as column >= 0; the most significant digit in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 2, 2, 2, 1, 4, 1, 1, 0, 1, 1, 3, 4, 4, 3, 1, 4, 1, 1, 0, 1, 1, 3, 1, 1, 6, 3, 1, 5, 1, 1, 0, 1, 1, 4, 1, 1, 1, 8, 4, 1, 5, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 5, 2, 6, 1, 1, 0, 1, 2, 1, 2, 3, 3, 3, 2, 1, 6, 2, 6, 1, 1, 0

Offset: 0

Antti Karttunen, Dec 19 2015

Square array A(row,col) is read by ascending antidiagonals as: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...

A265609(n,k) is the rising factorial, also known as Pochhammer symbol and A099563(n) is the most significant "digit" (place holder) in the factorial representation (A007623) of n.

			The top left corner of the array A265609 with its terms shown in factorial base (A007623) looks like this:
1,   0,    0,     0,       0,        0,         0,          0,           0
1,   1,   10,   100,    1000,    10000,    100000,    1000000,    10000000
1,  10,  100,  1000,   10000,   100000,   1000000,   10000000,   100000000
1,  11,  200,  2200,   30000,   330000,   4000000,   44000000,   500000000
1,  20,  310, 10000,  110000,  1220000,  14000000,  160000000,  1830000000
1,  21, 1100, 13300,  220000,  3000000,  36000000,  452000000,  5500000000
1, 100, 1300, 24000,  411000,  6000000,  82000000, 1100000000, 13300000000
1, 101, 2110, 41000, 1000000, 13000000, 174000000, 2374000000, 30360000000
-
Taking the most significant "digit" (placeholder that may get arbitrarily large values) gives us the top left corner of this array:
-
1, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1, 1, 2, 2, 3, 3, 4, 4,  5, 5,  6,  6,  7,  7,  8,  8,  9,  9, 10, 10, 11
1, 2, 3, 1, 1, 1, 1, 1,  1, 2,  2,  2,  2,  2,  2,  3,  3,  3,  3,  3,  3
1, 2, 1, 1, 2, 3, 3, 4,  5, 6,  7,  8, 10, 11, 12, 14, 15, 17, 19, 21,  1
1, 1, 1, 2, 4, 6, 8, 1,  1, 1,  1,  2,  2,  2,  2,  3,  3,  3,  4,  4,  5
1, 1, 2, 4, 1, 1, 1, 2,  3, 3,  4,  5,  6,  8,  9, 11, 12, 14, 16, 19, 21
1, 1, 3, 1, 1, 2, 3, 4,  6, 8, 11, 14,  1,  1,  1,  1,  2,  2,  2,  3,  3
1, 1, 3, 1, 2, 3, 5, 8,  1, 1,  1,  2,  2,  3,  4,  5,  6,  7,  8, 10, 12
1, 1, 4, 1, 3, 5, 9, 1,  2, 2,  3,  5,  6,  8, 11, 14, 17, 21,  1,  1,  1
1, 1, 1, 2, 4, 8, 1, 2,  3, 5,  7, 10, 14,  1,  1,  1,  2,  2,  3,  3,  4
1, 2, 1, 3, 6, 1, 2, 4,  6, 9, 14,  1,  1,  2,  3,  4,  5,  6,  8, 10, 13
1, 2, 1, 3, 1, 2, 3, 6, 10, 1,  1,  2,  3,  5,  6,  9, 12, 16, 21,  1,  1
1, 2, 1, 4, 1, 2, 5, 9,  1, 2,  3,  4,  7, 10, 14, 20,  1,  1,  2,  2,  3
1, 2, 2, 5, 1, 3, 7, 1,  2, 3,  5,  8, 13,  1,  1,  1,  2,  3,  4,  6,  8
...

Cf. A007623, A265609.
Column 1: A099563.
Row 0: A000007, rows 1 & 2: A000012, row 3: A008619 (see comment in A001710).
Row 4: 1,2,3 followed by A097992 ?
Main diagonal: A265891 (essentially, without the initial 1 from the corner of this array).
Cf. also array A265892.

(define (A265890 n) (A265890bi (A025581 n) (A002262 n)))
(define (A265890bi row col) (A099563 (A265609bi row col))) ;; Code for A265609bi given in A265609.

A265892 Array read by ascending antidiagonals: A(n,k) = A265893(A265609(n,k)), with n as row >= 0, k as column >= 0; the number of significant digits counted without trailing zeros in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 2, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 3, 2, 3, 2, 2, 1, 1, 0, 1, 2, 3, 2, 2, 3, 1, 1, 1, 0, 1, 3, 1, 2, 3, 1, 2, 2, 1, 1, 0, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 3, 3, 4, 2, 2, 2, 3, 3, 2, 1, 1, 0, 1, 1, 3, 2, 3, 3, 3, 2, 2, 1, 1, 1, 1, 0, 1, 3, 3, 4, 3, 4, 4, 4, 3, 3, 2, 2, 1, 1, 0

Offset: 0

Antti Karttunen, Dec 20 2015

Square array A(row,col) is read by ascending antidiagonals as: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...

			The top left corner of the array A265609 with its terms shown in factorial base (A007623) looks like this:
1,   0,    0,     0,       0,        0,         0,          0,           0
1,   1,   10,   100,    1000,    10000,    100000,    1000000,    10000000
1,  10,  100,  1000,   10000,   100000,   1000000,   10000000,   100000000
1,  11,  200,  2200,   30000,   330000,   4000000,   44000000,   500000000
1,  20,  310, 10000,  110000,  1220000,  14000000,  160000000,  1830000000
1,  21, 1100, 13300,  220000,  3000000,  36000000,  452000000,  5500000000
1, 100, 1300, 24000,  411000,  6000000,  82000000, 1100000000, 13300000000
1, 101, 2110, 41000, 1000000, 13000000, 174000000, 2374000000, 30360000000
-
Counting such digits for each term, but without the trailing zeros gives us the top left corner of this array:
-
The top left corner of the array:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1
1, 1, 2, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 1, 2, 3, 2
1, 2, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 3, 4, 3
1, 1, 2, 2, 3, 1, 2, 2, 3, 4, 3, 1, 2, 3, 4, 2, 3, 2, 3, 4, 1, 2, 3, 3, 4
1, 3, 3, 2, 1, 2, 3, 4, 4, 4, 3, 4, 2, 3, 3, 4, 3, 4, 3, 3, 4, 2, 4, 5, 4
1, 2, 1, 1, 2, 3, 4, 3, 3, 2, 3, 2, 4, 5, 4, 3, 4, 3, 3, 4, 5, 3, 4, 3, 4
1, 3, 2, 4, 3, 4, 3, 4, 2, 3, 4, 5, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 3
1, 2, 3, 2, 3, 4, 3, 4, 5, 3, 4, 5, 3, 4, 5, 3, 4, 5, 5, 4, 5, 4, 5, 3, 4
1, 3, 3, 4, 4, 4, 3, 4, 4, 5, 4, 3, 3, 5, 6, 6, 5, 6, 5, 6, 5, 6, 4, 5, 6
1, 1, 3, 3, 3, 2, 3, 3, 4, 4, 5, 3, 4, 5, 5, 4, 5, 4, 5, 4, 5, 6, 4, 5, 4
1, 3, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 4, 5, 6, 6, 5, 6, 5, 7, 6, 5, 5, 5, 5
1, 2, 3, 2, 4, 3, 4, 4, 4, 4, 5, 5, 6, 5, 5, 4, 6, 5, 6, 5, 4, 4, 4, 5, 6
1, 3, 1, 2, 3, 4, 5, 4, 3, 4, 4, 5, 5, 7, 6, 7, 6, 7, 5, 6, 7, 5, 4, 5, 6
1, 2, 4, 3, 5, 4, 3, 5, 6, 6, 5, 6, 6, 5, 6, 5, 6, 4, 5, 6, 4, 4, 6, 7, 8
1, 3, 3, 5, 4, 5, 5, 6, 5, 6, 5, 7, 6, 7, 6, 7, 4, 5, 6, 8, 5, 6, 7, 8, 6
1, 1, 3, 3, 4, 3, 5, 4, 5, 4, 6, 5, 6, 5, 6, 6, 7, 6, 7, 4, 5, 6, 7, 5, 6
...

Cf. A007623, A265609.
Row 0: A000007, rows 1-2: A000012, row 3: A000034 (see comment in A001710).
Column 0: A000012, column 1: A265893.
Cf. also array A265890.

(define (A265892 n) (A265892bi (A025581 n) (A002262 n)))
(define (A265892bi row col) (A265893 (A265609bi row col)))

A(n,k) = A265893(A265609(n,k)).

A265610 a(n) = rf(n, n+2)/(n+2)! - rf(n, n)/n!, rf the rising factorial A265609.

Original entry on oeis.org

-1, 0, 2, 11, 49, 204, 825, 3289, 13013, 51272, 201552, 791350, 3105322, 12183560, 47805615, 187623765, 736618125, 2893125840, 11367801060, 44686512090, 175739405790, 691437981000, 2721606268290, 10717182330426, 42219554975874, 166386610183024, 655976895434000

Offset: 0

Peter Luschny, Dec 19 2015

sign

Cf. A002054, A088218.

[Binomial(2*n+1, n-1)-(0^n + Binomial(2*n, n))/2: n in [0..30]]; // Vincenzo Librandi, Dec 20 2015

Join[{-1}, Table[Binomial[2 n + 1, n - 1] - Binomial[2 n, n]/2, {n, 1, 36}]] (* Vincenzo Librandi, Dec 20 2015 *)

A265610 = lambda n: rising_factorial(n, n+2)/factorial(n+2) - rising_factorial(n, n)/factorial(n)
print([A265610(n) for n in srange(27)])

G.f.: (-x^2+x-1-(x^2+3*x-1)/sqrt(1-4*x))/(2*x^2).
a(n) = binomial(2*n+1, n-1)-(0^n + binomial(2*n, n))/2 = A002054(n) - A088218(n).
a(n) = (1/2)*(3*n+2)*(n+2)*(n^2-1)*Gamma(2*n+1)/Gamma(n+3)^2 for n>=1.
a(n) ~ 4^n*(3/2)/sqrt(n*Pi).

A001710 Order of alternating group A_n, or number of even permutations of n letters.

Original entry on oeis.org

1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000

Offset: 0

N. J. A. Sloane

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003
a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003
Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004
Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004
The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008
Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008
First column of A092582. - Mats Granvik, Feb 08 2009
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009
For n>1: a(n) = A173333(n,2). - Reinhard Zumkeller, Feb 19 2010
Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010
For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011.
The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011
Also [1, 1] together with the row sums of triangle A162608. - Omar E. Pol, Mar 09 2012
a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11,  n=3: 112, n=4: 1123, 1132, 1213,  n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012
For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014
In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015
Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017
Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018
Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018
For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019
Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019
a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020
For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020
For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020

			G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 20. (a), c_n^e(t=1).
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
Camille Combe and Samuele Giraudo, Cliff operads: a hierarchy of operads on words, arXiv:2106.14552 [math.CO], 2021.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 7.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Shirali Kadyrov and Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Xah Lee, Combinatorics: Loop in n points.
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.
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]
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.
S-Z Song, S-G Hwang, S-H Rim, and G-S Cheon, Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Circular Permutation.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Even Permutation.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Odd Permutation.
Eric Weisstein's World of Mathematics, Transposition Graph.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index entries for sequences related to groups.
Index to divisibility sequences.

Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655.
a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
Bisections are A002674 and A085990 (essentially).
Row 3 of A265609 (essentially).
Row sums of A307429.

[1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014

seq(mul(k, k=3..n), n=0..20); # Zerinvary Lajos, Sep 14 2007

a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
a[n_]:= If[n<3, 1, n*a[n-1]]; Array[a, 21, 0]; (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2-x^2)/(2-2x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[-Log[1-x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
Numerator[Range[0, 20]!/2] (* Eric W. Weisstein, May 21 2017 *)
Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)

PARI
```
{a(n) = if( n<2, n>=0, n!/2)};
```

a(n)=polcoeff(1+x*sum(m=0,n,m^m*x^m/(1+m*x+x*O(x^n))^m),n) \\ Paul D. Hanna

A001710=n->n!\2+(n<2) \\ M. F. Hasler, Dec 01 2013

from math import factorial
def A001710(n): return factorial(n)>>1 if n > 1 else 1 # Chai Wah Wu, Feb 14 2023

def A001710(n): return (factorial(n) +int(n<2))//2
[A001710(n) for n in range(31)] # G. C. Greubel, Sep 28 2024

;; Using memoization-macro definec for which an implementation can be found in http://oeis.org/wiki/Memoization
(definec (A001710 n) (cond ((<= n 2) 1) (else (* n (A001710 (- n 1))))))
;; Antti Karttunen, Dec 19 2015

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
D-finite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) for n>2. - Chad Brewbaker, Jan 31 2003 [Corrected by N. J. A. Sloane, Jul 25 2008]
a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n-1} k*a(k). - Amarnath Murthy, Oct 29 2002
Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...]. - Michael Somos, Mar 04 2004
First Eulerian transform of A000027. See A000142 for definition of FET. - Ross La Haye, Feb 14 2005
From Paul Barry, Apr 18 2005: (Start)
a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/2)^2.
T(n,k) = abs(A008276(n, k)). (End)
E.g.f.: (2 - x^2)/(2 - 2*x).
E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.
E.g.f.: 1 + sinh(log(1/(1-x))). - Geoffrey Critzer, Dec 12 2010
a(n+1) = (-1)^n * A136656(n,1), n>=1.
a(n) = n!/2 for n>=2 (proof from the e.g.f). - Wolfdieter Lang, Apr 30 2010
a(n) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number (A000217). - Gary Detlefs, May 21 2010
a(n) = ( A000254(n) - 2* A001711(n-3) )/3, n>2. - Gary Detlefs, May 24 2010
O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. - Paul D. Hanna, Sep 13 2011
a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n). - Peter Luschny, Nov 07 2011
a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - Mircea Merca, Apr 07 2012
a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang. - Wolfdieter Lang, Jun 26 2012
G.f.: A(x) = 1 + x + x^2/(G(0)-2*x) where G(k) = 1 - (k+1)*x/(1 - x*(k+3)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012.
G.f.: 1 + x + (Q(0)-1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + (x*Q(x)-x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x/2 + (Q(0)-1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: 1+x + x^2*W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
From Antti Karttunen, Dec 19 2015: (Start)
a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]
For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End)
Inverse Stirling transform of a(n) is (-1)^(n-1)*A009566(n). - Anton Zakharov, Aug 07 2016
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/exp(n). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A006595(n-1)*n/A000124(n) for n>=2. - Anton Zakharov, Aug 23 2016
a(n) = A001563(n-1) - A001286(n-1) for n>=2. - Anton Zakharov, Sep 23 2016
From Peter Bala, May 24 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x - 1)*A(x) + 1 - x^2 = 0.
G.f.: A(x) = 1 + x + x^2/(1 - 3*x/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1  - ... - (n + 2)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1 + x + x^2/(1 - 2*x - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-2)) / 2 = x - 3*x^2/2! + 12*x^3/3! - ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1 - 2*x)^(-1/2) - 1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (-1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 2*(e-1).
Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001

Further terms from Simone Severini, Oct 15 2004

A226513 Array read by antidiagonals: T(n,k) = number of barred preferential arrangements of k things with n bars (k >=0, n >= 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 8, 13, 1, 4, 15, 44, 75, 1, 5, 24, 99, 308, 541, 1, 6, 35, 184, 807, 2612, 4683, 1, 7, 48, 305, 1704, 7803, 25988, 47293, 1, 8, 63, 468, 3155, 18424, 87135, 296564, 545835, 1, 9, 80, 679, 5340, 37625, 227304, 1102419, 3816548, 7087261

Offset: 0

N. J. A. Sloane, Jun 13 2013

The terms of this sequence are also called high-order Fubini numbers (see p. 255 in Komatsu). - Stefano Spezia, Dec 06 2020

			Array begins:
  1  1   3   13    75    541     4683     47293     545835 ...
  1  2   8   44   308   2612    25988    296564    3816548 ...
  1  3  15   99   807   7803    87135   1102419   15575127 ...
  1  4  24  184  1704  18424   227304   3147064   48278184 ...
  1  5  35  305  3155  37625   507035   7608305  125687555 ...
  1  6  48  468  5340  69516  1014348  16372908  289366860 ...
  ...
Triangle begins:
  1,
  1, 1,
  1, 2, 3,
  1, 3, 8, 13,
  1, 4, 15, 44, 75,
  1, 5, 24, 99, 308, 541,
  1, 6, 35, 184, 807, 2612, 4683,
  1, 7, 48, 305, 1704, 7803, 25988, 47293,
  1, 8, 63, 468, 3155, 18424, 87135, 296564, 545835
  ........
[_Vincenzo Librandi_, Jun 18 2013]

Z.-R. Li, Computational formulae for generalized mth order Bell numbers and generalized mth order ordered Bell numbers (in Chinese), J. Shandong Univ. Nat. Sci. 42 (2007), 59-63.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
Toka Diagana and Hamadoun Maïga, Some new identities and congruences for Fubini numbers, J. Number Theory 173 (2017), 547-569.
Takao Komatsu, Shifted Bernoulli numbers and shifted Fubini numbers, Linear and Nonlinear Analysis, Volume 6, Number 2, 2020, 245-263 (p. 255).

Rows 0-5 give: A000670, A005649, A226515, A226738, A226739, A226740.
Columns 2, 3 = A005563, A226514.
Cf. A053492 (array diagonal), A265609, A346982.

T:= (n, k)-> k!*coeff(series(1/(2-exp(x))^(n+1), x, k+1), x, k):
seq(seq(T(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Mar 26 2016

T[n_, k_] := Sum[StirlingS2[k, i]*i!*Binomial[n+i, i], {i, 0, k}]; Table[ T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 26 2016 *)

T(n,k) = Sum_{i=0..k} S2_k(i)*i!*binomial(n+i,i), where S2_k(i) is the Stirling number of the second kind. - Jean-François Alcover, Mar 26 2016
T(n,k) = k! * [x^k] 1/(2-exp(x))^(n+1). - Alois P. Heinz, Mar 26 2016
Conjectural g.f. for row n as a continued fraction of Stieltjes type: 1/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 4*x/(1 - (n+3)*x/(1 - 6*x/(1 - ... ))))))). Cf. A265609. - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
T(n,0) = 1; T(n,k) = Sum_{j=1..k} (n*j/k + 1) * binomial(k,j) * T(n,k-j).
T(n,0) = 1; T(n,k) = (n+1)*T(n,k-1) - 2*Sum_{j=1..k-1} (-1)^j * binomial(k-1,j) * T(n,k-j). (End)
G.f. for row n: (1/n!) * Sum_{m>=0} (n+m)! * x^m / Product_{j=1..m} (1 - j*x), for n >= 0. - Paul D. Hanna, Feb 01 2024

A303489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 8, 60, 1, 1, 10, 105, 840, 1, 1, 12, 162, 1920, 15120, 1, 1, 14, 231, 3640, 45045, 332640, 1, 1, 16, 312, 6144, 104720, 1290240, 8648640, 1, 1, 18, 405, 9576, 208845, 3674160, 43648605, 259459200, 1, 1, 20, 510, 14080, 375000, 8648640, 152152000, 1703116800, 8821612800

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

			Square array begins:
      1,      1,       1,       1,       1,       1,  ...
      1,      1,       1,       1,       1,       1,  ...
      6,      8,      10,      12,      14,      16,  ...
     60,    105,     162,     231,     312,     405,  ...
    840,   1920,    3640,    6144,    9576,   14080,  ...
  15120,  45045,  104720,  208845,  375000,  623645,  ...
=========================================================
A(1,1) = 1;
A(2,1) = 2*3 = 6;
A(3,1) = 3*4*5 = 60;
A(4,1) = 4*5*6*7 = 840;
A(5,1) = 5*6*7*8*9 = 15120, etc.
...
A(1,2) = 1;
A(2,2) = 2*4 = 8;
A(3,2) = 3*5*7 = 105;
A(4,2) = 4*6*8*10 = 1920;
A(5,2) = 5*7*9*11*13 = 45045, etc.
...
A(1,3) = 1;
A(2,3) = 2*5 = 10;
A(3,3) = 3*6*9 = 162;
A(4,3) = 4*7*10*13 = 3640;
A(5,3) = 5*8*11*14*17 = 104720, etc.
...

Index entries for sequences related to factorial numbers

Columns k=1..5 give A000407, A113551, A303486, A303487, A303488.
Main diagonal gives A061711.
Cf. A008279, A131182, A256268, A265609.

Table[Function[k, n! SeriesCoefficient[1/(1 - k x)^(n/k), {x, 0, n}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, Product[k i + n, {i, 0, n - 1}]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, k^n Pochhammer[n/k, n]][j - n + 1], {j, 0, 9}, {n, 0, j}] // Flatten

A(n,k) = Product_{j=0..n-1} (k*j + n).

A293475 a(n) = (3n + 4)Pochhammer(n, 4) / 4!.

Original entry on oeis.org

0, 7, 50, 195, 560, 1330, 2772, 5250, 9240, 15345, 24310, 37037, 54600, 78260, 109480, 149940, 201552, 266475, 347130, 446215, 566720, 711942, 885500, 1091350, 1333800, 1617525, 1947582, 2329425, 2768920, 3272360, 3846480, 4498472, 5236000, 6067215, 7000770

Offset: 0

Peter Luschny, Oct 20 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A265609, A293476, A293608, A293615.

[0] cat [(3*n + 4)*Factorial(n+3)/(Factorial(n-1)*Factorial(4)): n in [1..30]]; // G. C. Greubel, Nov 20 2017

A293475 := n -> (3*n + 4)*pochhammer(n, 4)/4!:
seq(A293475(n), n=0..32);

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 7, 50, 195, 560, 1330}, 32]
Table[n*StirlingS2[n+3, n+1], {n, 0, 50}] (* G. C. Greubel, Nov 20 2017 *)
f[n_] := (3n + 4) Pochhammer[n, 4]/4!; Array[f, 35, 0] (* or *)
CoefficientList[ Series[ x (7 + 8x)/(1 - x)^6, {x, 0, 34}], x] (* Robert G. Wilson v, Nov 21 2017 *)

for(n=0,30, print1(n*Stirling(n+3, n+1, 2), ", ")) \\ G. C. Greubel, Nov 20 2017

concat(0, Vec(x*(7 + 8*x) / (1 - x)^6 + O(x^40))) \\ Colin Barker, Nov 21 2017

a(n) = n*Stirling2(3 + n, 1 + n).
-a(-n-3) = (n + 3)*abs(Stirling1(n+2, n)) for n >= 0.
-a(-n-3) = a(n) + binomial(n+3, 4) for n >= 0.
From Colin Barker, Nov 21 2017: (Start)
G.f.: x*(7 + 8*x) / (1 - x)^6.
a(n) = n*(24 + 62*n + 57*n^2 + 22*n^3 + 3*n^4)/24.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 27*sqrt(3)*Pi/10 + 243*log(3)/10 - 2473/60.
Sum_{n>=1} (-1)^(n+1)/a(n) = 27*sqrt(3)*Pi/5 + 112*log(2)/5 - 2687/60. (End)

A293476 a(n) = ((n + 1)/2)(n + 2)Pochhammer(n, 5) / 4!.

Original entry on oeis.org

0, 15, 180, 1050, 4200, 13230, 35280, 83160, 178200, 353925, 660660, 1171170, 1987440, 3248700, 5140800, 7907040, 11860560, 17398395, 25017300, 35331450, 49092120, 67209450, 90776400, 121095000, 159705000, 208415025, 269336340, 344919330, 437992800, 551806200

Offset: 0

Peter Luschny, Oct 20 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A265609, A293475, A293608, A293615.

[0] cat [((n + 1)/2)*(n + 2)*Factorial(n+4)/(Factorial(4)*Factorial(n-1)): n in [1..30]]; // G. C. Greubel, Nov 20 2017

A293476 := n -> ((n+1)/2)*(n+2)*pochhammer(n, 5)/4!:
seq(A293476(n), n=0..11);

LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 15, 180, 1050, 4200, 13230, 35280, 83160}, 32]
Table[n*StirlingS2[4 + n, 1 + n], {n,0,50}] (* G. C. Greubel, Nov 20 2017 *)

for(n=0, 30, print1(n*stirling(n+4, n+1, 2), ", ")) \\ G. C. Greubel, Nov 20 2017

concat(0, Vec(15*x*(1 + 4*x + 2*x^2) / (1 - x)^8 + O(x^40))) \\ Colin Barker, Nov 21 2017

a(n) = n*Stirling2(4 + n, 1 + n).
-a(-n-4) = (n+4)*abs(Stirling1(n+3, n)) for n >= 0.
-a(-n-4) = a(n) + 5*binomial(n+4, 5)*(n+2) for n >= 0.
From Colin Barker, Nov 21 2017: (Start)
G.f.: 15*x*(1 + 4*x + 2*x^2) / (1 - x)^8.
a(n) = (1/48)*(n*(2 + 3*n + n^2)^2*(12 + 7*n + n^2)).
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>7. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 1187/36 - 10*Pi^2/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 653/36 - Pi^2/3 - 64*log(2)/3. (End)

A293608 a(n) = (3n + 7)Pochhammer(n, 5) / 4!.

Original entry on oeis.org

0, 50, 390, 1680, 5320, 13860, 31500, 64680, 122760, 218790, 370370, 600600, 939120, 1423240, 2099160, 3023280, 4263600, 5901210, 8031870, 10767680, 14238840, 18595500, 24009700, 30677400, 38820600, 48689550, 60565050, 74760840, 91626080, 111547920, 134954160

Offset: 0

Peter Luschny, Oct 20 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A265609, A293475, A293476, A293615.

[0] cat [(3*n + 7)*Factorial(n+4)/(Factorial(4)*Factorial(n-1)): n in [1..30]]; // G. C. Greubel, Nov 20 2017

A293608 := n -> (3*n+7)*pochhammer(n, 5)/4!:
seq(A293608(n), n=0..11);

LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 50, 390, 1680, 5320, 13860, 31500}, 32]
Table[n*(n+1)*StirlingS2[4 + n, 2 + n], {n,0,50}] (* G. C. Greubel, Nov 20 2017 *)
f[n_] := (3 n + 7) Pochhammer[n, 5]/4!; Array[f, 31, 0] (* or *)
CoefficientList[ Series[10x (5 + 4x)/(1 - x)^7, {x, 0, 30}], x] (* Robert G. Wilson v, Nov 21 2017 *)

for(n=0,30, print1(n*(n+1)*stirling(4 + n, 2 + n, 2), ", ")) \\ G. C. Greubel, Nov 20 2017

concat(0, Vec(10*x*(5 + 4*x) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Nov 21 2017

a(n) = n*(n+1)*Stirling2(4 + n, 2 + n).
-a(-n-4) = a(n) - 30*binomial(n+4, 5)*(n + 2) for n >= 0.
From Colin Barker, Nov 21 2017: (Start)
G.f.: 10*x*(5 + 4*x) / (1 - x)^7.
a(n) = (1/24)*(n*(168 + 422*n + 395*n^2 + 175*n^3 + 37*n^4 + 3*n^5)).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 8703/490 - 81*sqrt(3)*Pi/70 - 729*log(3)/70.
Sum_{n>=1} (-1)^(n+1)/a(n) = 81*sqrt(3)*Pi/35 + 416*log(2)/35 - 15298/735. (End)

A293615 a(n) = Pochhammer(n, 5) / 2.

Original entry on oeis.org

0, 60, 360, 1260, 3360, 7560, 15120, 27720, 47520, 77220, 120120, 180180, 262080, 371280, 514080, 697680, 930240, 1220940, 1580040, 2018940, 2550240, 3187800, 3946800, 4843800, 5896800, 7125300, 8550360, 10194660, 12082560, 14240160, 16695360, 19477920, 22619520

Offset: 0

Peter Luschny, Oct 20 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000389, A265609, A293475, A293476, A293608.

[0] cat [Factorial(n+4)/(2*Factorial(n-1)): n in [1..30]]; // G. C. Greubel, Nov 20 2017

A293615 := n -> pochhammer(n, 5)/2:
seq(A293615(n), n=0..11);

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 60, 360, 1260, 3360, 7560}, 32]
Table[Pochhammer[n, 5]/2, {n,0,50}] (* G. C. Greubel, Nov 20 2017 *)

for(n=0,30, print1(n*(n+1)*(n+2)*stirling(4 + n, 3 + n, 2), ", ")) \\ G. C. Greubel, Nov 20 2017

concat(0, Vec(60*x / (1 - x)^6 + O(x^40))) \\ Colin Barker, Nov 21 2017

a(n) = n*(n+1)*(n+2)*Stirling2(4 + n, 3 + n).
-a(-n-4) = a(n) for n >= 0.
a(n) = 60*A000389(n+4). - G. C. Greubel, Nov 20 2017
From Colin Barker, Nov 21 2017: (Start)
G.f.: 60*x / (1 - x)^6.
a(n) = (1/2)*(n*(1 + n)*(2 + n)*(3 + n)*(4 + n)).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5. (End)
From Amiram Eldar, Aug 31 2025: (Start)
Sum_{n>=1} 1/a(n) = 1/48.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/3 - 131/144. (End)

cp's OEIS Frontend

A265890 Array read by ascending antidiagonals: A(n,k) = A099563(A265609(n,k)), with n as row >= 0, k as column >= 0; the most significant digit in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

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A265892 Array read by ascending antidiagonals: A(n,k) = A265893(A265609(n,k)), with n as row >= 0, k as column >= 0; the number of significant digits counted without trailing zeros in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

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A265610 a(n) = rf(n, n+2)/(n+2)! - rf(n, n)/n!, rf the rising factorial A265609.

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A001710 Order of alternating group A_n, or number of even permutations of n letters.

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PARI

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A226513 Array read by antidiagonals: T(n,k) = number of barred preferential arrangements of k things with n bars (k >=0, n >= 0).

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A303489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals: A(n,k) = n! * [x^n] 1/(1 - k*x)^(n/k).

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A293475 a(n) = (3*n + 4)*Pochhammer(n, 4) / 4!.

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A293475 a(n) = (3n + 4)Pochhammer(n, 4) / 4!.

A293476 a(n) = ((n + 1)/2)(n + 2)Pochhammer(n, 5) / 4!.

A293608 a(n) = (3n + 7)Pochhammer(n, 5) / 4!.