A186430 -id:A186430 - cp's OEIS frontend

A053657 a(n) = Product_{p prime} p^{ Sum_{k>=0} floor[(n-1)/((p-1)p^k)]}.

1, 2, 24, 48, 5760, 11520, 2903040, 5806080, 1393459200, 2786918400, 367873228800, 735746457600, 24103053950976000, 48206107901952000, 578473294823424000, 1156946589646848000, 9440684171518279680000, 18881368343036559360000, 271211974879377138647040000

Offset: 1

Jean-Luc Chabert, Feb 16 2000

LCM of denominators of the coefficients of x^n*z^k in {-log(1-x)/x}^z as k=0..n, as described by triangle A075264.
Denominators of integer-valued polynomials on prime numbers (with degree n): 1/a(n) is a generator of the ideal formed by the leading coefficients of integer-valued polynomials on prime numbers with degree less than or equal to n.
Also the least common multiple of the orders of all finite subgroups of GL_n(Q) [Minkowski]. Schur's notation for the sequence is M_n = a(n+1). - Martin Lorenz (lorenz(AT)math.temple.edu), May 18 2005
This sequence also occurs in algebraic topology where it gives the denominators of the Laurent polynomials forming a regular basis for K*K, the hopf algebroid of stable cooperations for complex K-theory. Several different equivalent formulas for the terms of the sequence occur in the literature. An early reference is K. Johnson, Illinois J. Math. 28(1), 1984, pp.57-63 where it occurs in lines 1-5, page 58. A summary of some of the other formulas is given in the appendix to K. Johnson, Jour. of K-theory 2(1), 2008, 123-145. - Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008
a(n) is divisible by n!, by Legendre's formula for the highest power of a prime that divides n!. Also, a(n) is divisible by (n+1)! if and only if n+1 is not prime. - Jonathan Sondow, Jul 23 2009
Triangle A163940 is related to the divergent series 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ... for m =>-1. The left hand columns of this triangle can be generated with the MC polynomials, see A163972. The Minkowski numbers appear in the denominators of these polynomials. - Johannes W. Meijer, Oct 16 2009
Unsigned Stirling numbers of the first kind as [s + k, k] (Karamata's notation) where k = {0, 1, 2, ...} and s is in general complex results in Pochhammer[s,k]*(integer coefficient polynomial of (k-1) degree in s) / M[k], where M[k] is the least common multiple of the orders of all finite groups of n x n-matrices over rational numbers (Minkowiski's theorem) which is sequence A053657. - Lorenz H. Menke, Jr., Feb 02 2010
From Peter Bala, Feb 21 2011: (Start)
Given a subset S of the integers Z, Bhargava has shown how to associate with S a generalized factorial function, denoted n!_S, which shares many properties of the classical factorial function n!.
The present sequence is the generalized factorial function n!S associated with the set of primes S = {2,3,5,7,...}. The associated generalized exponential function E(x) = Sum{n>=1} x^(n-1)/a(n) vanishes at x = -2: i.e. Sum_{n>=1} (-2)^n/a(n) = 0.
For the table of associated generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) see A186430.
This sequence is related to the Bernoulli polynomials in two ways [Chabert and Cahen]:
(1) a(n) = (n-1)!*A001898(n-1).
(2) (t/(exp(t)-1))^x = sum {n = 0..inf} P(n,x)*t^n/a(n+1),
where the P(n,x) are primitive polynomials in the ring Z[x].
If p_1,...,p_n are any n primes then the product of their pairwise differences Product_{i(End)
LCM of denominators of the coefficients of S(m+n-1,m) as polynomial in m of degree 2*(n-1), as described by triangle A202339. - Vladimir Shevelev, Dec 17 2011
Sometimes called "Minkowski numbers" (e.g., by Guralnick and Lorenz), after the German mathematician Hermann Minkowski (1864-1909). - Amiram Eldar, Aug 24 2024

			a(7)=24^3*Product_{i=1..3} A202318(i)=24^3*1*10*21=2903040. - _Vladimir Shevelev_, Dec 17 2011

Jean-Luc Chabert, Scott T. Chapman, and William W. Smith, A basis for the ring of polynomials integer-valued on prime numbers, in: Daniel Anderson (ed.), Factorization in integral domains, Lecture Notes in Pure and Appl. Math. 189, Dekker, New York, 1997.

Gheorghe Coserea, Table of n, a(n) for n = 1..541
F. Bencherif, Sur une propriété des polynômes de Stirling, 26th Journées Arithmétiques, July 6-10, 2009, Université Jean Monnet, Saint-Etienne, France. [From _Jonathan Sondow_, Jul 23 2009]
M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (2000), 783-799.
Paul-Jean Cahen, and J. L. Chabert, What You Should Know About Integer-Valued Polynomials, The American Mathematical Monthly, 123 (No. 4, 2016), 311-337.
J.-L. Chabert, Integer-valued polynomials on prime numbers and logarithm power expansion, European J. Combinatorics 28 (2007) 754-761. [From _Jonathan Sondow_, Jul 23 2009]
J. L. Chabert, About polynomials whose divided differences are integer-valued on prime numbers, ICM 2012 Proceedings, vol. I, pp. 1-7. Complete proceedings. (warning: file size is 26MB). - From _N. J. A. Sloane_, Nov 28 2012
J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences.
Robert M. Guralnick and Martin Lorenz, Orders of Finite Groups of Matrices, in: William Chin, James Osterburg and Declan Quinn (eds.), Groups, Rings and Algebras, Contemporary Mathematics, Vol. 420 (2006), pp. 141-161; arXiv preprint, arXiv:math/0511191 [math.GR], 2005; author's link.
K. Johnson, The action of the stable operations of complex K-theory on coefficient groups, Illinois J. Math. 28(1), 1984, pp. 57-63. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
K. Johnson, The invariant subalgebra and anti-invariant submodule of K_*K_{(p)}, Jour. of K-theory 2(1), 2008, 123-145. [From Keith Johnson (johnson(AT)mscs.dal.ca), Nov 03 2008]
Hermann Minkowski, Zur Theorie der quadratischen Formen, J. Reine Angew. Math. 101 (1887), 196-202. ( = Ges. Abh., pp. 212-218, Chelsea, New York, 1967.)
Issai Schur, Über eine Klasse von endlichen Gruppen linearer Substitutionen, Sitzungsber. Preuss. Akad. Wiss. (1905), 77-91. ( = Ges. Abh., Bd. 1, pp. 128-142, Springer-Verlag, Berlin-Heidelberg-New York, 1973.)
J.-P. Serre, Bounds for the orders of the finite subgroups of G(k), Group Representation Theory (eds. M. Geck, D. Testerman, J. Thevenaz), EPFL Press, Lausanne, 2006, 405-450.
Wikipedia, Bhargava factorial.
Gregory Gerard Wojnar, Daniel Sz. Wojnar, and Leon Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017. See p. 14.

Cf. A002207, A075264, A075266, A075267.
a(n) = n!*A163176(n). - Jonathan Sondow, Jul 23 2009
Cf. A202318.
Appears in A163972. - Johannes W. Meijer, Oct 16 2009
Cf. A001898, A007814, A163940, A186430, A202339, A212429, A346046.

A053657 := proc(n) local P,p,q,s,r;
P := select(isprime,[$2..n]); r:=1;
for p in P do s := 0; q := p-1;
do if q > (n-1) then break fi;
s := s + iquo(n-1,q); q := q*p; od;
r := r * p^s; od; r end: # Peter Luschny, Jul 26 2009
ser := series((y/(exp(y)-1))^x, y, 20): a := n -> denom(coeff(ser, y, n-1)):
seq(a(n), n=1..19); # Peter Luschny, May 13 2019

m = 16; s = Expand[Normal[Series[(-Log[1-x]/x)^z, {x, 0, m}]]];
a[n_, k_] := Denominator[ Coefficient[s, x^n*z^k]];
Prepend[Apply[LCM, Table[a[n,k], {n,m}, {k,n}], {1}], 1]
(* Jean-François Alcover, May 31 2011 *)
a[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[ Range[n] ]}]; Array[a, 30] (* Jean-François Alcover, Nov 22 2016 *)

{a(n)=local(X=x+x^2*O(x^n),D);D=1;for(j=0,n-1,D=lcm(D,denominator( polcoeff(polcoeff((-log(1-X)/x)^z+z*O(z^j),j,z),n-1,x))));return(D)} /* Paul D. Hanna, Jun 27 2005 */

{a(n)=prod(i=1,#factor(n!)~,prime(i)^sum(k=0,#binary(n), floor((n-1)/((prime(i)-1)*prime(i)^k))))} /* Paul D. Hanna, Jun 27 2005 */

S(n, p) = {
     my(acc = 0, tmp = p-1);
     while (tmp < n, acc += floor((n-1)/tmp); tmp *= p);
     return(acc);
};
a(n) = {
     my(rv = 1);
     forprime(p = 2, n, rv *= p^S(n,p));
     return(rv);
};
vector(17, i, a(i))  \\ Gheorghe Coserea, Aug 24 2015

a(2n) = 2*a(2n-1). - Jonathan Sondow, Jul 23 2009
a(2*n+1) = 24^n * Product_{i=1..n} A202318(i). - Vladimir Shevelev, Dec 17 2011
For n>=0, A007814(a(n+1)) = n+A007814(n!). - Vladimir Shevelev, Dec 28 2011
a(n) = denominator([y^(n-1)] (y/(exp(y)-1))^x). - Peter Luschny, May 13 2019
Sum_{n>=1} 1/a(n) = A346046. - Amiram Eldar, Jul 02 2023

More terms from Paul D. Hanna, Jun 27 2005

A202917 For n >= 0, let n!^(1) = A053657(n+1) and, for 0 <= m <= n, C^(1)(n,m) = n!^(1)/(m!^(1)*(n-m)!^(1)). The sequence gives a triangle of numbers C^(1)(n,m) with rows of length n+1.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 60, 10, 60, 1, 1, 1, 10, 10, 1, 1, 1, 126, 21, 1260, 21, 126, 1, 1, 1, 21, 21, 21, 21, 1, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Dec 26 2011

1) Note that A053657(n+1) is the LCM of the denominators of the coefficients of the polynomials Q^(1)n(x) which, for integer x=k, are defined by the recursion Q^(1)_0(x)=1, for n>=1, Q^(1)_n(x) = Sum{i=1..k} i*Q^(1)(n-1)(i). Also note that Q^(1)_n(k) = S(k+n,k), where the numbers S(l,m) are Stirling numbers of the second kind. The sequence of polynomials {Q^(1)_n(x)} includes the family of sequences of polynomials {{Q^(r)_n}}(r>=0) described in a comment at A175669. In particular, the LCM of the denominators of the coefficients of Q^(0)_n(x) is n!.
2) This triangle differs from triangle A186430 which is defined according to the theory of factorials over sets by Bhargava. Unfortunately, this theory does not have a conversion theorem. Therefore it is not known if there is a set A such that n!^(1) = n!_A in the Bhargava sense.
3) If p is an odd prime, then the (p-1)-th row contains two 1's and p-2 numbers that are multiples of p. For a conjectural generalization, see comment in A175669.

			Triangle begins
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....6.....1
.3..|..1.....1 ... 1  .....1
.4..|..1....60....10......60.....1
.5..|..1.....1....10......10.....1.....1
.6..|..1...126....21....1260....21...126.....1
.7..|..1.....1....21......21....21....21.....1.....1
.8..|

Cf. A175669, A053657, A202339, A202367, A202368, A202369

A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; f1[n_] := A053657[n+1]; C1[n_, m_] := f1[n]/(f1[m] * f1[n-m]); Table[C1[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2016 *)

A007814(C^(1)(n,m)) = A007814(C(n,m)).

A186432 Triangle associated with the set S of squares {0,1,4,9,16,...}.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 30, 30, 1, 1, 56, 140, 56, 1, 1, 90, 420, 420, 90, 1, 1, 132, 990, 1848, 990, 132, 1, 1, 182, 2002, 6006, 6006, 2002, 182, 1, 1, 240, 3640, 16016, 25740, 16016, 3640, 240, 1, 1, 306, 6120, 37128, 87516, 87516, 37128, 6120, 306, 1, 1, 380, 9690, 77520, 251940, 369512, 251940, 77520, 9690, 380, 1

Offset: 0

Peter Bala, Feb 22 2011

Given a subset S of the integers Z, Bhargava [1] has shown how to associate with S a generalized factorial function, denoted n!_S, sharing many properties of the classical factorial function n! (which corresponds to the choice S = Z). In particular, he shows that the generalized binomial coefficients n!_S/(k!_S*(n-k)!_S) are always integral for any choice of S. Here we take S = {0,1,4,9,16,...}, the set of squares.
The associated generalized factorial function n!_S is given by the formula
n!S = Product{k=0..n} (n^2 - k^2), with the convention 0!S = 1. This should be compared with n! = Product{k=0..n} (n - k).
For n >= 1, n!_S = (2*n)!/2 = A002674(n).
Compare this triangle with A086645 and also A186430 - the generalized binomial coefficients for the set S of prime numbers {2,3,5,7,11,...}.

			Triangle begins
n/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1....12.....1
.3..|..1....30....30.....1
.4..|..1....56...140....56.....1
.5..|..1....90...420...420....90.....1
.6..|..1...132...990..1848...990...132.....1
.7..|..1...182..2002..6006..6006..2002...182.....1
...

M. Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (2000), 783-799.

Cf. A002114, A086645, A186430, A186433 (inverse).

Table[2 Binomial[2 n, 2 k] - Boole[Or[k == 0, k == n]], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 23 2017 *)

TABLE ENTRIES
T(n,k) = n!_S/(k!_S*(n-k)!_S),
which simplifies to
T(n,k) = 2*binomial(2*n,2*k) for 1 <= k < n,
with boundary conditions T(n,0) = 1 and T(n,n) = 1 for n >= 0.
RELATIONS WITH OTHER SEQUENCES
Denote this triangle by T. The first column of the inverse T^-1 (see A186433) begins [1, -1, 11, -301, 15371, ...] and, apart from the initial 1, is a signed version of the Glaisher's H' numbers A002114.
The first column of (1/2)*T^2 begins [1/2, 1, 7, 31, 127, ...] and, apart from the initial term, equals A000225(2*n-1), counting the preferential arrangements on (2*n - 1) labeled elements having less than or equal to two ranks.
The first column of (1/3)*T^3 begins [1/3, 1, 13, 181, 1933, ...] and, apart from the initial term, is A101052(2*n-1), which gives the number of preferential arrangements on (2*n-1) labeled elements having less than or equal to three ranks.

A204087 Reduced Pascal triangle: C_R(n,m) = A003418(n) / max(A003418(m), A003418(n-m)), m=0,...,n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 6, 2, 1, 1, 5, 10, 10, 5, 1, 1, 1, 5, 10, 5, 1, 1, 1, 7, 7, 35, 35, 7, 7, 1, 1, 2, 14, 14, 70, 14, 14, 2, 1, 1, 3, 6, 42, 42, 42, 42, 6, 3, 1, 1, 1, 3, 6, 42, 42, 42, 6, 3, 1, 1, 1, 11, 11, 33, 66, 462, 462, 66, 33, 11, 11, 1

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 10 2012

The sixth row is the first one which differs from triangles A080381, A080396.

			Triangle begins:
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....2.....1
.3..|..1.....3.....3.....1
.4..|..1.....2.....6.....2.....1
.5..|..1.....5....10....10.....5.....1
.6..|..1.....1.....5....10.....5.....1.....1
.7..|..1.....7.....7....35....35.....7.....7.....1

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A007318, A080381, A080396, A186430, A202917, A202941.

g:= proc(n) option remember; `if`(n=0, 1, ilcm(g(n-1), n)) end:
CR:= proc(n, m) option remember; g(n)/max(g(m), g(n-m)) end:
seq (seq (CR(n,m), m=0..n), n=0..11); # Alois P. Heinz, Jan 11 2012

g[n_] := g[n] = If[n == 0, 1, LCM[g[n-1], n]]; CR[n_, m_] := CR[n, m] = g[n]/Max[ g[m], g[n-m]]; Table[Table[CR[n, m], {m, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *)

A204088 Row sums of A204087.

Original entry on oeis.org

1, 2, 4, 8, 12, 32, 24, 100, 132, 188, 148, 1168, 708, 3200, 2344, 1488, 2120, 21456, 16596, 222948, 130572, 38196, 29800, 492248, 299096, 529712, 455424, 1143404, 920536, 20232316, 13769088, 226481600, 252603072, 52242944, 40457056, 28671168, 16885280

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jan 10 2012

nonn

If n is prime, then a(n)==2 mod n. Up to n=200, there are only two composite numbers with such a property: a(49)=623017500436==2 mod 49 and a(51)=391496243228==2 mod 51.

Cf. A204087, A186430, A202917, A202941, A203269, A203278.

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cp's OEIS Frontend

A186431 Row sums of A186430.

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A053657 a(n) = Product_{p prime} p^{ Sum_{k>=0} floor[(n-1)/((p-1)p^k)]}.

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A202917 For n >= 0, let n!^(1) = A053657(n+1) and, for 0 <= m <= n, C^(1)(n,m) = n!^(1)/(m!^(1)*(n-m)!^(1)). The sequence gives a triangle of numbers C^(1)(n,m) with rows of length n+1.

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A186432 Triangle associated with the set S of squares {0,1,4,9,16,...}.

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A204087 Reduced Pascal triangle: C_R(n,m) = A003418(n) / max(A003418(m), A003418(n-m)), m=0,...,n.

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A204088 Row sums of A204087.

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