A164652 -id:A164652 - cp's OEIS frontend

A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 4, 2, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 4, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 4, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 4, 2, 8, 4, 4, 4, 4, 4, 4, 2, 4, 4, 4, 2, 8, 2, 4, 8

Offset: 1

N. J. A. Sloane

If n = Product p_i^a_i, d = Product p_i^c_i is a unitary divisor of n if each c_i is 0 or a_i.
Also the number of squarefree divisors of n. - Labos Elemer
Also number of divisors of the squarefree kernel of n: a(n) = A000005(A007947(n)). - Reinhard Zumkeller, Jul 19 2002
Also shadow transform of pronic numbers A002378.
For n >= 1 define an n X n (0,1) matrix A by A[i,j] = 1 if lcm(i,j) = n, A[i,j] = 0 if lcm(i,j) <> n for 1 <= i,j <= n. a(n) is the rank of A. - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 11 2003
a(n) is also the number of solutions to x^2 - x == 0 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
a(n) is the number of squarefree divisors of n, but in general the set of unitary divisors of n is not the set of squarefree divisors (compare the rows of A077610 and A206778). - Jaroslav Krizek, May 04 2009
Row lengths of the triangles in A077610 and in A206778. - Reinhard Zumkeller, Feb 12 2012
a(n) is also the number of distinct residues of k^phi(n) (mod n), k=0..n-1. - Michel Lagneau, Nov 15 2012
a(n) is the number of irreducible fractions y/x that satisfy x*y=n (and gcd(x,y)=1), x and y positive integers. - Luc Rousseau, Jul 09 2017
a(n) is the number of (x,y) lattice points satisfying both x*y=n and (x,y) is visible from (0,0), x and y positive integers. - Luc Rousseau, Jul 10 2017
Conjecture: For any nonnegative integer k and positive integer n, the sum of the k-th powers of the unitary divisors of n is divisible by the sum of the k-th powers of the odd unitary divisors of n (note that this sequence lists the sum of the 0th powers of the unitary divisors of n). - Ivan N. Ianakiev, Feb 18 2018
a(n) is the number of one-digit numbers, k, when written in base n such that k and k^2 end in the same digit. - Matthew Scroggs, Jun 01 2018
Dirichlet convolution of A271102 and A000005. - Vaclav Kotesovec, Apr 08 2019
Conjecture: Let b(i; n), n > 0, be multiplicative sequences for some fixed integer i >= 0 with b(i; p^e) = (Sum_{k=1..i+1} A164652(i, k) * e^(k-1)) * (i+2) / (i!) for prime p and e > 0. Then we have Dirichlet generating functions: Sum_{n > 0} b(i; n) / n^s = (zeta(s))^(i+2) / zeta((i+2) * s). Examples for i=0 this sequence, for i=1 A226602, and for i=2 A286779. - Werner Schulte, Feb 17 2022
The smallest integer with 2^m unitary divisors, or equivalently, the smallest integer with 2^m squarefree divisors, is A002110(m). - Bernard Schott, Oct 04 2022

			a(12) = 4 because the four unitary divisors of 12 are 1, 3, 4, 12, and also because the four squarefree divisors of 12 are 1, 2, 3, 6.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.

T. D. Noe, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 49-50.
Lorenz Halbeisen, A number-theoretic conjecture and its implication for set theory, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.
Jon Maiga, Upper bound of Fibonacci entry points, 2019.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Eric Weisstein's World of Mathematics, Unitary Divisor Function.
Wikipedia, Unitary divisor.
Index entries for sequences computed from exponents in factorization of n

Cf. A077610, A048105, A000173, A013928, A000079, A001221, A002110, A034448, A047994, A061142, A277561.
Sum of the k-th powers of the squarefree divisors of n for k=0..10: this sequence (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: this sequence (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), this sequence (k=10).
Cf. A020821 (Dgf at s=2), A177057 (Dgf at s=4).

a034444 = length . a077610_row  -- Reinhard Zumkeller, Feb 12 2012

[#[d:d in Divisors(n)|Gcd(d,n div d) eq 1]:n in [1..110]]; // Marius A. Burtea, Jan 11 2020

[&+[Abs(MoebiusMu(d)):d in Divisors(n)]:n in [1..110]]; // Marius A. Burtea, Jan 11 2020

with(numtheory): for n from 1 to 200 do printf(`%d,`,2^nops(ifactors(n)[2])) od:
with(numtheory);
# returns the number of unitary divisors of n and a list of them
f:=proc(n)
local ct,i,t1,ans;
ct:=0; ans:=[];
t1:=divisors(n);
for i from 1 to nops(t1) do
d:=t1[i];
if igcd(d,n/d)=1 then ct:=ct+1; ans:=[op(ans),d]; fi;
od:
RETURN([ct,ans]);
end;
# N. J. A. Sloane, May 01 2013
# alternative Maple program:
a:= n-> 2^nops(ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Jan 23 2024
a := n -> 2^NumberTheory:-NumberOfPrimeFactors(n, distinct):  # Peter Luschny, May 13 2025

a[n_] := Count[Divisors[n], d_ /; GCD[d, n/d] == 1]; a /@ Range[105] (* Jean-François Alcover, Apr 05 2011 *)
Table[2^PrimeNu[n],{n,110}] (* Harvey P. Dale, Jul 14 2011 *)

a(n)=1<Charles R Greathouse IV, Feb 11 2011

for(n=1, 100, print1(direuler(p=2, n, (1+X)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Sep 26 2020

from sympy import divisors, gcd
def a(n):
    return sum(1 for d in divisors(n) if gcd(d, n//d)==1)
# Indranil Ghosh, Apr 16 2017

from sympy import primefactors
def a(n): return 2**len(primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 16 2017

(define (A034444 n) (if (= 1 n) n (* 2 (A034444 (A028234 n))))) ;; Antti Karttunen, May 29 2017

a(n) = Sum_{d|n} abs(mu(n)) = 2^(number of different primes dividing n) = 2^A001221(n), with mu(n) = A008683(n). [Added Möbius formula. - Wolfdieter Lang, Jan 11 2020]
a(n) = Product_{ primes p|n } (1 + Legendre(1, p)).
Multiplicative with a(p^k)=2 for p prime and k>0. - Henry Bottomley, Oct 25 2001
a(n) = Sum_{d|n} tau(d^2)*mu(n/d), Dirichlet convolution of A048691 and A008683. - Benoit Cloitre, Oct 03 2002
Dirichlet generating function: zeta(s)^2/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
Inverse Mobius transform of A008966. - Franklin T. Adams-Watters, Sep 11 2005
Asymptotically [Finch] the cumulative sum of a(n) = Sum_{n=1..N} a(n) ~ (6/(Pi^2))*N*log(N) + (6/(Pi^2))*(2*gamma - 1 - (12/(Pi^2))*zeta'(2))*N + O(sqrt(N)). - Jonathan Vos Post, May 08 2005 [typo corrected by Vaclav Kotesovec, Sep 13 2018]
a(n) = Sum_{d|n} floor(rad(d)/d), where rad is A007947 and floor(rad(n)/n) = A008966(n). - Enrique Pérez Herrero, Nov 13 2009
a(n) = A000005(n) - A048105(n); number of nonzero terms in row n of table A225817. - Reinhard Zumkeller, Jul 30 2013
G.f.: Sum_{n>0} A008966(n)*x^n/(1-x^n). - Mircea Merca, Feb 25 2014
a(n) = Sum_{d|n} lambda(d)*mu(d), where lambda is A008836. - Enrique Pérez Herrero, Apr 27 2014
a(n) = A277561(A156552(n)). - Antti Karttunen, May 29 2017
a(n) = A005361(n^2)/A005361(n). - Velin Yanev, Jul 26 2017
L.g.f.: -log(Product_{k>=1} (1 - mu(k)^2*x^k)^(1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = Sum_{d|n} A001615(d) * A023900(n/d). - Torlach Rush, Jan 20 2020
Sum_{d|n, gcd(d, n/d) = 1} a(d) * (-1)^omega(n/d) = 1. - Amiram Eldar, May 29 2020
a(n) = lim_{k->oo} A000005(n^(2*k))/A000005(n^k). - Velin Yanev and Amiram Eldar, Jan 10 2025

More terms from James Sellers, Jun 20 2000

A105751 Imaginary part of Product_{k=0..n} (1 + k*i), i = sqrt(-1).

Original entry on oeis.org

0, 1, 3, 0, -40, -90, 1050, 6160, -46800, -549900, 3103100, 67610400, -271627200, -11186357000, 26495469000, 2416003824000, -1394099824000, -662595375078000, -936096296850000, 225382826562400000, 819329864480400000, -93217812901913700000, -570263312237604700000

Offset: 0

Paul Barry, Apr 18 2005

From Peter Bala, Jun 01 2023: (Start)
Compare with A105750(n) = the real part of Product_{k = 0..n} (1 + k*sqrt(-1)). Moll (2012) studied the prime divisors of the terms of A105750 and divided the primes into three classes. Numerical calculation suggests that a similar division holds in this case.
Type 1: primes p that do not divide any element of the sequence {a(n)}.
In this case, unlike in A105750, the set of type 1 primes is empty; that is, every prime p divides some term of this sequence.
Type 2: primes p such that the p-adic valuation v_p(a(n)) has asymptotically linear behavior. An example is given below.
We conjecture that the set of type 2 primes consists of primes p == 1 (mod 4), equivalently, rational primes that split in the field extension Q(sqrt(-1)) of Q, together with the prime p = 2, which ramifies in Q(sqrt(-1)). See A002144.
Moll's conjecture 5.5 extends to this sequence and takes the form:
(i) the 2-adic valuation v_2(a(n)) ~ n/4 as n -> oo.
(ii) for the other primes of type 2, the p-adic valuation v_p(a(n)) ~ n/(p - 1) as n -> oo.
Type 3: primes p such that the sequence of p-adic valuations {v_p(a(n)) : n >= 0} exhibits an oscillatory behavior (this phrase is not precisely defined). An example is given below.
We conjecture that the set of type 3 primes consists of primes p == 3 (mod 4), equivalently, rational primes that remain inert in the field extension Q(sqrt(-1)) of Q. See A002145. (End)

			From _Peter Bala_, Jun 01 2023: (Start)
The sequence of 5-adic valuations [v_5(a(n)) : n = 4..100] = [1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 12, 11, 11, 13, 11, 12, 13, 13, 12, 12, 14, 13, 13, 14, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 18, 18, 18, 18, 18, 20, 19, 19, 20, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 24, 25, 25, 24, 24, 25, 25, 25].
Note that v_5(a(100)) = 25 = 100/(5 - 1), in agreement with the asymptotic behavior conjectured above.
The sequence of 3-adic valuations [v_3(a(n)) : n >= 4] begins [0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 3, 1, 0, 3, 3, 0, 1, 3, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 2, 1, 0, 2, 2, 0, 1, 2, 0, 3, ...], exhibiting the oscillatory behavior for type 3 primes conjectured above. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A003703, A009454, A048994, A105750, A164652, A231531, A363409 - A363416.

a:= proc(n) option remember; `if`(n<2, n,
      ((2*n-1)*a(n-1)-(n^2-2*n+2)*n*a(n-2))/(n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 11 2018

Table[Im[Product[1+k*I,{k,0,n}]],{n,0,22}] (* James C. McMahon, Jan 27 2024 *)

a(n) = imag(prod(k=0, n, 1+k*I)); \\ Michel Marcus, Apr 11 2018

from sympy.functions.combinatorial.numbers import stirling
def A105751(n): return sum(stirling(n+1,n-(k<<1),kind=1)*(-1 if k&1 else 1) for k in range((n>>1)+1)) # Chai Wah Wu, Feb 22 2024

a(n) = ((2*n-1)*a(n-1)-(n^2-2*n+2)*n*a(n-2))/(n-1) for n > 1, a(n) = n for n < 2. - Alois P. Heinz, Apr 11 2018
From Peter Bala, May 27 2023:(Start)
a(n) = Sum_{k = 0..floor((n+1)/2)} (-1)^k*|Stirling1(n+1, n-2*k)|, where Stirling1(n, k) = A048994(n,k).
The triangular number n*(n+1)/2 divides a(n). See A164652. In particular, if p is an odd prime then p divides a(p).
a(2*n) = (-1)^(n+1)*A003703(2*n+1) for n >= 0.
a(2*n+1) = (-1)^(n+1)*A009454(2*n+2) for n >= 0. (End)

A189507 Triangle read by rows: T(n,k) (n >= 0, 1 <= k <= n+1) are the signed Hultman numbers.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 20, 21, 6, 1, 148, 160, 65, 10, 1, 1348, 1620, 701, 155, 15, 1, 15104, 19068, 9324, 2247, 315, 21, 1, 198144, 264420, 138016, 38029, 5908, 574, 28, 1, 2998656, 4166880, 2325740, 692088, 124029, 13524, 966, 36, 1, 51290496, 74011488, 43448940, 13945700, 2723469, 344961, 27930, 1530, 45, 1, 979732224, 1459381440, 897020784, 305142068, 64711856, 8996295, 850905, 53262, 2310, 55, 1

Offset: 0

N. J. A. Sloane, Apr 23 2011

"Signed" refers to the fact that these numbers are associated with signed permutations. The numbers themselves are positive.

			Triangle begins:
      1
      1     1
      4     3    1
     20    21    6    1
    148   160   65   10   1
   1348  1620  701  155  15  1
  15104 19068 9324 2247 315 21 1
  ...

N. Alexeev, A. Pologova, M. A. Alekseyev, Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs, Journal of Computational Biology 24:2 (2017), 93-105. doi:10.1089/cmb.2016.0190 arXiv:1503.05285
Simona Grusea and Anthony Labarre, The distribution of cycles in breakpoint graphs of signed permutations, arXiv:1104.3353v1

The first three columns give A001171, A189508, A189509. Cf. A164652.

A185263 Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t) related to Hultman numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 15, 8, 1, 35, 84, 1, 70, 469, 180, 1, 126, 1869, 3044, 1, 210, 5985, 26060, 8064, 1, 330, 16401, 152900, 193248, 1, 495, 39963, 696905, 2286636, 604800, 1, 715, 88803, 2641925, 18128396, 19056960, 1, 1001, 183183, 8691683, 109425316, 292271616, 68428800, 1, 1365, 355355, 25537655, 539651112, 2961802480, 2699672832

Offset: 0

N. J. A. Sloane, Jan 21 2012

Row n contains floor(n/2) + 1 terms.

			Triangle begins:
  n\k| 0    1      2       3         4         5        6
-----+---------------------------------------------------
   0 | 1
   1 | 1
   2 | 1    1
   3 | 1    5
   4 | 1   15      8
   5 | 1   35     84
   6 | 1   70    469     180
   7 | 1  126   1869    3044
   8 | 1  210   5985   26060      8064
   9 | 1  330  16401  152900    193248
  10 | 1  495  39963  696905   2286636    604800
  11 | 1  715  88803 2641925  18128396  19056960
  12 | 1 1001 183183 8691683 109425316 292271616 68428800
  ...
Polynomials p_n(t):
  p_0 = t;
  p_1 = t^2;
  p_2 = t^3 +     t;
  p_3 = t^4 +   5*t^2;
  p_4 = t^5 +  15*t^3 +    8*t;
  p_5 = t^6 +  35*t^4 +   84*t^2;
  p_6 = t^7 +  70*t^5 +  469*t^3 +  180*t;
  p_7 = t^8 + 126*t^6 + 1869*t^4 + 3044*t^2;
  ...
A(x;t) = t + t^2*x/1! + (t^3 + t)*x^2/2! + (t^4 + 5*t^2)*x^3/3! + ...

Gheorghe Coserea, Rows n = 0..202, flattened
Nikita Alexeev and Peter Zograf, Hultman numbers, polygon gluings and matrix integrals, arXiv preprint arXiv:1111.3061 [math.PR], 2011.

Cf. A038720, A048994, A060593, A185259.

For uncompressed form of polynomial coefficients, in order of increasing powers, see A164652.

T[n_, k_] := Abs[StirlingS1[n+2, n-2k+1]]/Binomial[n+2, 2];
Table[T[n, k], {n, 0, 13}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

seq(N) = {
  my(p=vector(N), t='t, v); p[1] = t^2; p[2] = t^3 + t;
  for (n=3, N,
    p[n] = ((2*n+1)*t*p[n-1] + (n-1)*(n^2-t^2)*p[n-2])/(n+2));
  v = vector(#p, n, vector(1+n\2, k, polcoeff(p[n], n+1-2*(k-1))));
  concat([[1]], v);
};
concat(seq(13))

N=14; x='x+O('x^(N+1));
concat(apply(p->select(a->a!=0, Vec(p)), Vec(serlaplace(((1-x)^(-t) - (1+x)^t)/x^2))))

T(n,k) = -stirling(n+2, n+1-2*k, 1)/binomial(n+2,2);
concat(1, concat(vector(13, n, vector(1+n\2, k, T(n, k-1)))))
\\ Gheorghe Coserea, Jan 29 2018

From Gheorghe Coserea, Jan 29 2018: (Start)
p(n) = Sum_{k=0..floor(n/2)} T(n,k)*t^(n+1-2*k) satisfies (n+2)*p(n) = (2*n+1)*t*p(n-1) + (n-1)*(n^2-t^2)*p(n-2), n >= 2. (th. 3, (iii))
E.g.f. A(x;t) = Sum_{n>=0} p(n)*x^n/n! = ((1-x)^(-t) - (1+x)^t)/x^2. (th. 3, (i))
T(n,k) = -Stirling1(n+2, n+1-2*k)/binomial(n+2,2), where Stirling1(n,k) is defined by A048994.
A000142(n) = p(n)(1), A052572(n) = p(n)(2) for n > 0, A060593(n) = T(2*n, n) for n > 0. (End)
n-th row polynomial R(n,x) satisfies x*R(n,x^2) = (1/2)*( P(n+1,x) - P(n+1,-x) )/ binomial(n+2,2), where P(k,x) = (1 + x)*(1 + 2*x) * ... *(1 + k*x). - Peter Bala, May 14 2023

More terms from Gheorghe Coserea, Jan 29 2018

A247500 Triangle read by rows: T(n, k) = n!*binomial(n + 1, k)/(k + 1)!, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 12, 6, 1, 24, 60, 40, 10, 1, 120, 360, 300, 100, 15, 1, 720, 2520, 2520, 1050, 210, 21, 1, 5040, 20160, 23520, 11760, 2940, 392, 28, 1, 40320, 181440, 241920, 141120, 42336, 7056, 672, 36, 1, 362880, 1814400, 2721600, 1814400, 635040, 127008, 15120, 1080, 45, 1

Offset: 0

Peter Luschny, Oct 17 2014

An alternative definition would have been: (n-k)!*N(n,k) where N(n,k) are the little Narayana numbers A090181(n,k). This adds a first column (1,0,0,...) to the triangle and amounts to (Gamma(n)*Gamma(n+1))/(Gamma(k)*Gamma(k+1)*Gamma(n-k+2)). - Peter Luschny, Jun 18 2015
From Peter Bala, Sep 03 2023: (Start)
Let E(y) = Sum_{n >= 0} y^n/(n+1)!. Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!*(n+1)! as defined in Wang and Wang.
Let B(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = 1/sqrt(y)*BesselI(1,2*sqrt(y)). A generating function for the triangle is E(y)*B(x*y) = 1 + (1 + x)*y/(1!*2!) + (2 + 3*x + x^2)*y^2/(2!*3!) + (6 + 12*x + 6*x^2 + x^3)*y^3/(3!*4!) + .... Cf. A105278 with a generating function exp(y)*B(x*y).
The n-th power of this array has a generating function E(y)^n*B(x*y). In particular, the matrix inverse has a generating function B(x*y)/E(y). (End)

			Triangle begins:
                      1;
                   1,    1;
                2,    3,    1;
             6,   12,    6,    1;
         24,   60,   40,   10,    1;
     120,  360,  300,  100,   15,    1;
  720, 2520, 2520, 1050,  210,   21,    1;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A247499 (row sums), A008297.
Cf. A000142, A000217, A001710.
Cf. A204515 (central terms), A105278, A004736.
Cf. A090181, A001263.

a247500 n k = a247500_tabl !! n !! k
a247500_row n = a247500_tabl !! n
a247500_tabl = zipWith (zipWith div) a105278_tabl a004736_tabl
-- Reinhard Zumkeller, Oct 19 2014

/* triangle */ [[Factorial(n)/Factorial(k) * Binomial(n+2, k+1) /(n+2): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 18 2014

T := (n,k) -> ((k+1)*(n+1)*GAMMA(n+1)^2)/(GAMMA(k+2)^2*GAMMA(n-k+2));
A247500 := (n, k) -> (n!/(k+1)!)*binomial(n + 1, k):

Table[((k + 1) (n + 1) Gamma[n + 1]^2)/(Gamma[k + 2]^2*
Gamma[n - k + 2]), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 19 2015 *)

T(n, k) = ((k+1)*(n+1)*Gamma(n+1)^2)/(Gamma(k+2)^2 *Gamma(n-k+2)). (original name)
T(n, k) = (n!/k!)*C(n+2, k+1)/(n+2).
T(n, 0) = A000142(n).
T(n, n-1) = A000217(n).
T(n+1, 1) = A001710(n+2).
Sum_{k=0..n} T(n, k) = A247499(n).
L(n+1, k+1) = T(n-1, k)*P(n) for n>=1 and 0<=k<=n; here L(n,k) denote the unsigned Lah numbers and P(n) the pronic numbers. - Peter Luschny, Oct 18 2014
T(n,k) = A105278(n+1,k+1) / (n+1-k), k=0..n. - Reinhard Zumkeller, Oct 19 2014
From Peter Bala, May 24 2023: (Start)
Triangle equals A164652 * A008277 (assuming the same offset for the three triangles).
This is equivalent to the Stirling number identity Sum_{i = 0..n} (n+1)!/(i+1)!* binomial(n,i)*Stirling1(i+1,k) = (-1)^(n+k+1)*Stirling1(n+1,k) for n, k >= 0. (End)

Name updated by Peter Luschny, Jan 09 2022

A264614 Irregular triangle read by rows: T(n,k) = number of unsigned unichromosonal genomes with n genes at 3-break distance k from a fixed genome, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 4, 1, 1, 10, 13, 1, 20, 75, 24, 1, 35, 287, 397, 1, 56, 854, 3112, 1017, 1, 84, 2142, 16196, 21897, 1, 120, 4746, 64520, 222573, 70920, 1, 165, 9570, 212498, 1486749, 1919817, 1, 220, 17919, 606584, 7503815, 24312636, 7475625, 1, 286, 31603, 1548404, 30891575, 200350670, 246179061

Offset: 1

N. J. A. Sloane, Nov 28 2015

			Triangle begins:
1,
1,0,
1,1,
1,4,1,
1,10,13,
1,20,75,24,
1,35,287,397,
1,56,854,3112,1017,
1,84,2142,16196,21897,
...

N. Alexeev, A. Pologova, M. A. Alekseyev, Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs, Journal of Computational Biology 24:2 (2017), 93-105. doi:10.1089/cmb.2016.0190 arXiv:1503.05285

Cf. A164652, A189507, A264615, A264616, A264617.

Extended and offset corrected by Max Alekseyev, Feb 13 2018

A264615 Irregular triangle read by rows: T(n,k) = number of signed unichromosonal genomes with n genes at 3-break distance k from a fixed genome, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 22, 25, 1, 50, 333, 1, 95, 1851, 1893, 1, 161, 6839, 39079, 1, 252, 19782, 323580, 301505, 1, 372, 48510, 1706180, 8566857, 1, 525, 105546, 6792650, 95942613, 82953225, 1, 715, 209682, 22248446, 668057885, 3025374471, 1, 946, 387783, 63055388, 3435912383, 43154349714, 35095900185

Offset: 1

N. J. A. Sloane, Nov 28 2015

			Triangle begins:
1,
1,1,
1,7,
1,22,25,
1,50,333,
1,95,1851,1893,
1,161,6839,39079,
1,252,19782,323580,301505,
1,372,48510,1706180,8566857,
...

N. Alexeev, A. Pologova, M. A. Alekseyev, Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs, Journal of Computational Biology 24:2 (2017), 93-105. doi:10.1089/cmb.2016.0190 arXiv:1503.05285

Cf. A164652, A189507, A264614, A264616, A264617.

Extended and offset corrected by Max Alekseyev, Feb 13 2018

A264616 Irregular triangle read by rows: T(n,k) = number of unsigned unichromosonal genomes with n genes at 4-break distance k from a fixed genome, 0 <= k <= floor((n+1)/3).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 5, 1, 15, 8, 1, 35, 84, 1, 70, 649, 1, 126, 3585, 1328, 1, 210, 14949, 25160, 1, 330, 50421, 312128, 1, 495, 144903, 2621465, 861936, 1, 715, 367983, 16015637, 23532464, 1, 1001, 847275, 76717355, 401435968, 1, 1365, 1801943, 304775471, 4519766436, 1400675584

Offset: 1

N. J. A. Sloane, Nov 28 2015

			Triangle begins:
1,
1,0,
1,1,
1,5,
1,15,8,
1,35,84,
1,70,649,
1,126,3585,1328,
1,210,14949,25160,
...

N. Alexeev, A. Pologova, M. A. Alekseyev, Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs, Journal of Computational Biology 24:2 (2017), 93-105. doi:10.1089/cmb.2016.0190 arXiv:1503.05285

Cf. A164652, A189507, A264614, A264615, A264617.

Extended and offset corrected by Max Alekseyev, Feb 13 2018

A264617 Irregular triangle read by rows: T(n,k) = number of signed unichromosonal genomes with n genes at 4-break distance k from a fixed genome, 0 <= k <= floor((n+1)/3).

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 47, 1, 175, 208, 1, 470, 3369, 1, 1036, 45043, 1, 2002, 315213, 327904, 1, 3522, 1472157, 8846240, 1, 5775, 5287071, 180501713, 1, 8965, 15795483, 1908769247, 1791317504, 1, 13321, 41169051, 13068136571, 68640287456, 1, 19097, 96558891, 66722214923, 1895171760688

Offset: 1

N. J. A. Sloane, Nov 28 2015

			Triangle begins:
1,
1,1,
1,7,
1,47,
1,175,208,
1,470,3369,
1,1036,45043,
1,2002,315213,327904,
1,3522,1472157,8846240,
...

N. Alexeev, A. Pologova, M. A. Alekseyev, Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs, Journal of Computational Biology 24:2 (2017), 93-105. doi:10.1089/cmb.2016.0190 arXiv:1503.05285

Cf. A164652, A189507, A264614, A264615, A264616.

Extended and offset corrected by Max Alekseyev, Feb 13 2018

A260695 a(n) is the number of permutations p of {1,..,n} such that the minimum number of block interchanges required to sort the permutation p to the identity permutation is maximized.

Original entry on oeis.org

1, 1, 1, 5, 8, 84, 180, 3044, 8064, 193248, 604800, 19056960, 68428800, 2699672832, 10897286400, 520105017600, 2324754432000, 130859579289600, 640237370572800, 41680704936960000, 221172909834240000, 16397141420298240000, 93666727314800640000, 7809290721329061888000, 47726800133326110720000

Offset: 0

Marion Scheepers, Nov 16 2015

nonn

Interweaving of nonzero Hultman numbers A164652(n,k) for k=1 and k=2. - Max Alekseyev, Nov 20 2020

			The next three lines illustrate applying block interchanges to [2 4 6 1 3 5 7], an element of S_7.
Step 1: [2 4 6 1 3 5 7]->[3 5 1 2 4 6 7]-interchange blocks 3 5 and 2 4 6.
Step 2: [3 5 1 2 4 6 7]->[4 1 2 3 5 6 7]-interchange blocks 3 5 and 4.
Step 3: [4 1 2 3 5 6 7]->[1 2 3 4 5 6 7]-interchange blocks 4 and 1 2 3.
As [2 4 6 1 3 5 7] requires 3 = floor(7/2) block interchanges, it is one of the a(7) = 3044.
Each of the 23 non-identity elements of S_4 requires at least 1 block interchange to sort to the identity. But only 8 of these require 2 block interchanges, the maximum number required for elements of S_4. They are: [4 3 2 1], [4 1 3 2], [4 2 1 3], [3 1 4 2], [3 2 4 1], [2 4 1 3], [2 1 4 3], [2 4 3 1]. So, a(4) = 8.

D. A. Christie, Sorting Permutations by Block-Interchanges, Inf. Process. Lett. 60 (1996), 165-169.
Robert Cori, Michel Marcus, and Gilles Schaeffer, Odd permutations are nicer than even ones, European Journal of Combinatorics 33:7 (2012), 1467-1478.
M. Tikhomirov, A conjecture harmonic numbers, MathOverflow, 2020.
D. Zagier, On the distribution of the number of cycles of elements in symmetric groups.

The number of elements of S_n that can be sorted by: a single block interchange (A145126), two block interchanges (A228401), three block interchanges (A256181), context directed block interchanges (A249165).
The number of signed permutations that can be sorted by: context directed reversals (A260511), applying either context directed reversals or context directed block interchanges (A260506).
Cf. A000254, A001008, A002805, A164652.

a[n_]:=Abs[StirlingS1[n+2,Mod[n,2]+1]/Binomial[n+2,2]]; Array[a,25,0] (* Stefano Spezia, Apr 01 2024 *)

{ A260695(n) = abs(stirling(n+2, n%2+1)) / binomial(n+2, 2); } \\ Max Alekseyev, Nov 20 2020

For even n, a(n) = 2 * n! / (n+2).
For odd n, a(n) = 2 * n! * H(n+1) / (n+2) = 2 * A000254(n+1) / ((n+1)*(n+2)), where H(n+1) = A001008(n+1)/A002805(n+1) is the (n+1)-st harmonic number.
a(n) =  A164652(n, 1+(n mod 2)). - Max Alekseyev, Nov 20 2020

Edited and extended by Max Alekseyev incorporating comments from M. Tikhomirov, Nov 20 2020

cp's OEIS Frontend

A034444 a(n) is the number of unitary divisors of n (d such that d divides n, gcd(d, n/d) = 1).

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A105751 Imaginary part of Product_{k=0..n} (1 + k*i), i = sqrt(-1).

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A189507 Triangle read by rows: T(n,k) (n >= 0, 1 <= k <= n+1) are the signed Hultman numbers.

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A185263 Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t) related to Hultman numbers.

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A247500 Triangle read by rows: T(n, k) = n!*binomial(n + 1, k)/(k + 1)!, 0 <= k <= n.

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A264614 Irregular triangle read by rows: T(n,k) = number of unsigned unichromosonal genomes with n genes at 3-break distance k from a fixed genome, 0 <= k <= floor(n/2).

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