Thomas Ward - cp's OEIS frontend

A341991 Multiplicative defect in a natural approximation for the terms of A341617.

1, 1, 1, 1, 2, 2, 1, 1, 4, 12, 6, 6, 6, 6, 3, 1, 8, 8, 12, 12, 6, 6

Offset: 1

Thomas Ward, Feb 25 2021

A coarse approximation to A341617(n) is the primorial of (n-1), and the terms of this sequence are the quotient A341617(n) divided by the primorial of (n-1).

			For n = 3 it is known that A341617(3) = 2, so a(3) = 2/(3-1)! = 1.

P. Miska and T. Ward, Stirling numbers and periodic points, arXiv:2102.07561 [math.NT], 2021.

Cf. A341617, A000142, A007947, A002110.

a(n) = A341617(n)/radical((n-1)!) = A341617(n)/(n-1)# = A341617(n)/A002110(n-1).

A341617 Repair factors for Stirling numbers of the second kind.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 30, 210, 840, 2520, 1260, 13860, 13860, 180180, 90090, 30030, 240240, 4084080, 6126120, 116396280, 58198140, 58198140

Offset: 1

Thomas Ward, Feb 16 2021

a(k) is the smallest number with the property that the sequence (a(k)*S(n+k-1, k))_{n>=1}, viewed as a sequence in n, counts the periodic points of some map. Here S(n, k) denotes the Stirling numbers of the second kind, so S(n, k) is the number of ways to partition a set of n elements into k nonempty subsets.
Because these numbers relate to properties of the Stirling number sequence S(n, k) viewed as a sequence in n, it is natural to think of these values as indexed by k.
The values generated by a finite calculation are inherently empirical, calculated as the least common multiple of the first 3000 terms and then checked for the repair property for period up to 50,000. For smaller values of k (those listed here) various ad hoc arguments can be used to check the empirical candidate values.
The empirical values can be calculated using a simple PARI code, and this only gives a possible candidate value for a(k). If that value happens to be (k-1)! then it is correct. If it is a strict divisor of (k-1)! then further checks are needed for each prime in the quotient (candidate value)/(k-1)!. The potentially unbounded nature of the calculation required would make a proof of a closed formula particularly interesting.
(EMPIRICAL PARI CODE TO GENERATE CANDIDATE VALUES) for(k=1, 40, a=vector(3000, n, stirling(n+k-1, k, 2)); b=vector(length(a), n, (1/n)*sumdiv(n, d, moebius(n/d)*a[d])); print1(denominator(b)", "))

			The statement that a(3) = 2 means that the sequence of Stirling numbers S_3 = (1, 6, 25, 90, ...) (that is, the sequence A000392 with an offset of 3) does not have the property of counting periodic points for some map, but does have this property after multiplication by 2 (which gives A028243 with an offset of 2), and 2 is the smallest integer with this property. This specific value is immediately known to be exact, because a(3) divides (3-1)! = 2.

P. Miska and T. Ward, Stirling numbers and periodic points, arXiv:2102.07561 [math.NT], 2021.

Cf. A000392, A028243, A008277, A341991.

It is known that a(k) divides (k-1)!.

There is an explicit formula for a(k), but it involves an in principle infinite calculation as follows: compute the set of rational numbers {(1/n) Sum_{d|n} mu(n/d)*S(d+k-1, k): n>=1}, and then define a(k) to be the least common multiple of the denominators of that (possibly infinite) set of rational numbers. Here mu denotes the classical Möbius function, and the sum is taken over the divisors d of n. Strictly speaking, any calculation only gives candidate values which are initially only known to be a factor of the real value, which in turn is a factor of (k-1)!. For small values of k listed above ad hoc arguments are needed to check the candidate values evaluated as the least common multiple of the denominators of the first 3000 terms.

A159282 Denominator of the rational coefficient in the main term in the dynamical analog of Mertens's theorem for a full n-dimensional shift, n >= 2.

Original entry on oeis.org

6, 12, 1620, 2160, 2551500, 3061800, 33756345000, 38578680000, 4060381958325000, 4511535509250000, 3168740859543387253125000, 3456808210410967912500000, 34159303730702924635072148437500

Offset: 2

Thomas Ward, Apr 08 2009

a(n) for n >= 2 may be defined as follows. For a full n-dimensional shift, let M(N) = Sum_{L} O(L)/exp(h[L]), where the sum is over subgroups L of finite index in Z^n, O(L) is the number of points with stabilizer L, and exp(h) is the number of symbols.

Then M(N) is asymptotic to a rational times a power of Pi times a product of values of the zeta function at odd integers, and a(n) is the denominator of that rational.

			For n = 3, using the formula in terms of residues, we have residue(zeta(z-1) * zeta(z) * zeta(z+1) * N^z/z, z=2) = (1/12) * zeta(3) * Pi^2 * N^2, so a(3) = 12 (and A159283(3) = 1). [Because A159283(n) = 1 for n = 2..11, these ten values are not listed in the OEIS.]

Vaclav Kotesovec, Table of n, a(n) for n = 2..63
R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.

This is the denominator of a rational sequence whose numerator is A159283.

# The following program generates an expression from which denominator a(n) can be read off:
f:=n->residue(product(Zeta(z-j),j=-1..(n-2))*N^z/z,z=n-1):
seq(f(n), n=2..30);

Denominator[Table[Residue[Product[Zeta[z - j], {j, -1, n-2}]/z, {z, n-1}], {n, 2, 14}]] (* Vaclav Kotesovec, Sep 05 2019 *)

By Perron's formula, M(N) = residue(zeta(z+1) * ... * zeta(z-n+2) * N^z, z=n-1) = (b(n)/a(n)) * N^(d-1) * Pi^(floor(n/2)*(floor(n/2)+1)) * Product_{j=1..floor((n-1)/2)} zeta(2*j+1), where b(n) = A159283(n).

Various sections edited by Petros Hadjicostas, Feb 20 2021

A159283 Numerator of the rational coefficient in the main term in the dynamical analog of Mertens's theorem for a full n-dimensional shift, n >= 12 (it is 1 for 2 <= n <= 11).

Original entry on oeis.org

691, 691, 691, 691, 2499347, 2499347, 109638854849, 109638854849, 19144150084038739, 19144150084038739, 1487175010978381361737, 1487175010978381361737, 351514769627820131218308186067

Offset: 12

Thomas Ward, Apr 08 2009

a(n) for n >= 2 may be defined as follows. For a full n-dimensional shift, let M(N) = Sum_{L} O(L)/exp(h[L]), where the sum is over subgroups L of finite index in Z^n, O(L) is the number of points with stabilizer L, and exp(h) is the number of symbols.

Then M(N) is asymptotic to a rational times a power of Pi times a product of values of the zeta function at odd integers, and a(n) is the numerator of that rational.

			For n = 12, using the formula in terms of residues, we have residue(zeta(z+1) * ... * zeta(z-10) * N^z/z, z=11) = (691/3168740859543387253125000) * zeta(3) * zeta(5) * zeta(7) * zeta(9) * zeta(11) * Pi^42 * N^11, so a(12) = 691 and A159282(12) = 3168740859543387253125000.

R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.

This is the numerator of a rational sequence whose denominator is A159282.

# The following program generates an expression from which numerator a(n) can be read off:
f:=n->residue(product(Zeta(z-j),j=-1..(n-2))*N^z/z,z=n-1):
seq(f(n), n=2..30);

Numerator[Table[Residue[Product[Zeta[z - j], {j, -1, n-2}]/z, {z, n-1}][[1]], {n, 12, 24}]] (* Vaclav Kotesovec, Sep 05 2019 *)

M(N) = residue(zeta(z+1) * ... * zeta(z-n+2) * N^z, z=n-1) = (a(n)/b(n)) * N^(d-1) * Pi^(floor(n/2)*(floor(n/2)+1)) * Product_{j=1..floor((n-1)/2)} zeta(2*j+1), where b(n) = A159282(n).

Various sections edited by Petros Hadjicostas, Feb 20 2021

A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4sin^2(Pij*x) dx.

Original entry on oeis.org

2, 4, 6, 10, 12, 20, 24, 34, 44, 64, 78, 116, 148, 208, 286, 410, 556, 808, 1120, 1620, 2308, 3352, 4784, 6980, 10064, 14680, 21296, 31128, 45276, 66288, 96712, 141654, 207156, 303716, 444748, 652612, 956884, 1404920, 2062080, 3029564, 4450120

Offset: 1

Thomas Ward, Jan 07 2008

nonn

This quantity arises in some examples associated to the dynamical Mertens's theorem for quasihyperbolic toral automorphisms.
The function being integrated to compute a_n vanishes on the set of points in the Farey sequence of level n. I am particularly interested in knowing how large the sequence is asymptotically.
a(n) = coefficient of x^(n*(n+1)/2) in the polynomial (-1)^n*Product_{k=1..n} (1-x^k)^2, and is the maximal such coefficient as well. - Steven Finch, Feb 03 2009

			a(2) = 4 since Integral_{0..1} sin^2(Pi*x) sin^2(2*Pi*x) dx = 1/4.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000 (terms 1..174 from Robert Israel)
Miklos Bóna, R. Gómez, M. D. Ward, Workshop in Analytic and Probabilistic Combinatorics BIRS-16w5048 2016.
S. R. Finch, Signum equations and extremal coefficients.
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
Jeffrey Gaither, Guy Louchard, Stephan Wagner, and Mark Daniel Ward, Resolution of T. Ward's Question and the Israel-Finch Conjecture: Precise Analysis of an Integer Sequence Arising in Dynamics, Combinatorics, Probability and Computing, 24 (2015), 195-215. Special Issue Honouring the Memory of Philippe Flajolet.
S. Jaidee, S. Stevens and T. Ward, Mertens' theorem for toral automorphisms, arXiv:0801.2082 [math.DS], 2008-2010.
S. Jaidee, S. Stevens and T. Ward, Mertens' theorem for toral automorphisms, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1819-1824.
Yasuhiko Kamiyama, The Euler characteristic of the fiber product of Morse functions, Bull. Korean Math. Soc. (2025) Vol. 62, No. 1, pp. 71-80. See pp. 73, 75.
Mark Daniel Ward, Resolution of T. Ward's Question and the Israel-Finch Conjecture. Precise Asymptotic Analysis of an Integer Sequence Motivated by the Dynamical Mertens' Theorem for Quasihyperbolic Toral Automorphisms, Slides, 2013.
T. Ward, D. W. Cantrell and R. Israel, sci.math.research discussion, 2008.

Cf. A005728, A047653.

a:= n->int(product(4*(sin(Pi*j*x))^2, j=1..n), x=0..1); seq(a(n), n=1..10);
# second Maple program:
A133871:= k -> (-1)^k*coeff(mul((t^j-1)^2,j=1..k),t,k*(k+1)/2);
# Robert Israel, Mar 15 2013

p = 1; Table[p = Expand[p*(1 - x^n)^2]; Max[(-1)^n*CoefficientList[p, x]], {n, 1, 100}] (* Vaclav Kotesovec, May 03 2018 *)
(* The constant "d" *) Chop[-E^(-I*(Pi^2*(1 + 6*x^2) - 6*PolyLog[2, E^(2*I*Pi*x)]) / (6*Pi*x)) /. x -> (x /. FindRoot[Pi*(Pi*(-1 + 6*x^2) + 12*I*x*Log[1 - E^(2*I*Pi*x)]) + 6*PolyLog[2, E^(2*I*Pi*x)], {x, 4/5}, WorkingPrecision -> 100])] (* Vaclav Kotesovec, May 04 2018 *)

a(n)=sum(k=0, n*(n+1)/2, polcoeff(prod(m=1, n, 1-x^m+x*O(x^k)), k)^2) \\ Paul D. Hanna

a(n) = sum of squares of coefficients in Product_{k=1..n} (1-x^k). - Paul D. Hanna, Nov 30 2010

a(n) ~ c * d^n / sqrt(n), where d = 1.48770584269062356180051131... and c = 2.40574583936181024... [Ward, 2013]. - Vaclav Kotesovec, May 03 2018

More terms from Steven Finch, Feb 03 2009

A091112 Number of orbits of length n under the map whose periodic points are counted by A061686.

Original entry on oeis.org

1, 8, 513, 115272, 70162625, 95640604266, 256797561193432, 1238094271228829120, 9993778343964199218438, 127849400250667505250954500, 2480163309080566931933236667234, 70354340598798824605743590305386600, 2830805474672999382519296750329811657242

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old Name was: "A061686 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n under that map".

			b(1)=1, b(3)=1540, so a(3)=(1/3)(b(3)-b(1))=513.

Robert Israel, Table of n, a(n) for n = 1..126
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Thomas Ward, Exactly realizable sequences. [local copy].

Cf. A061686.

a061686:= proc(n) option remember;
  add(binomial(n,k)^5*(n-k)*procname(k)/n, k=0..n-1)
end proc:
a061686(0):= 1:
a:= n -> 1/n * add(numtheory:-mobius(d)*a061686(n/d), d = numtheory:-divisors(n)):
seq(a(n), n=1..6); # Robert Israel, May 05 2015

(* b = A061686 *) b[0]=1; b[n_] := b[n] = Sum[Binomial[n, k]^5*(n-k)*b[k]/ n, {k, 0, n-1}]; a[n_] := (1/n)*DivisorSum[n, MoebiusMu[#] * b[n/#] &]; Array[a, 20] (* Jean-François Alcover, Dec 04 2015 *)

A091112(n)=sumdiv(n,d,moebius(d)*A061686(n/d)) \\ M. F. Hasler, May 11 2015

If b(n) is the (n+1)-th term of A061686, then a(n) = (1/n)*Sum_{d|n} mu(d) b(n/d).

More terms from Robert Israel, May 05 2015

Name clarified by M. F. Hasler, May 11 2015

A091160 Number of orbits of length n under the map whose periodic points are counted by A061687.

Original entry on oeis.org

1, 16, 2835, 2370752, 6611343125, 48887897438124, 821067869874486556, 28006755051982013513984, 1782755223314276717178818904, 198173677662343700104263938337400, 36467946245662764068249155883368682252, 10631160782054640951386529213624176084501136

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old name was: A061687 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n for that map.

			b(1)=1, b(3)=8506, so a(3) = (1/3)*(8506-1) = 2835.

Alois P. Heinz, Table of n, a(n) for n = 1..100
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Thomas Ward, Exactly realizable sequences. [local copy].

Cf. A061687.

with(numtheory):
b:= proc(n) option remember;
      `if`(n=0, 1, add(binomial(n, k)^6*(n-k)*b(k)/n, k=0..n-1))
    end:
a:= n-> add(mobius(d)*b(n/d), d=divisors(n))/n:
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 19 2014

b[n_] := b[n] = If[n==0, 1, Sum[Binomial[n, k]^6 (n-k)b[k]/n, {k, 0, n-1}]];
a[n_] := Sum[MoebiusMu[d] b[n/d], {d, Divisors[n]}]/n;
Array[a, 15] (* Jean-François Alcover, Nov 18 2020, after Alois P. Heinz *)

If b(n) is the (n+1)th term of A061687, then a(n) = (1/n)*Sum_{d|n} mu(d)*b(n/d).

More terms from Alois P. Heinz, Mar 19 2014

Name clarified by Michel Marcus, May 13 2015

A091201 Number of orbits of length n under the map whose periodic points are counted by A061688.

Original entry on oeis.org

1, 32, 16281, 52293792, 692825815625, 28927809504181734

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old name was: A061688 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n under that map.

			b(1)=1,b(3)=48844, so a(3)=(1/3)(48844-1)=16281.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Thomas Ward, Exactly realizable sequences. [local copy].

Cf. A061688.

If b(n) is the (n+1)th term of A061688, then a(n) = (1/n)*Sum_{d|n}mu(d)b(n/d).

Name clarified by Michel Marcus, May 14 2015

A091266 Number of orbits of length n under the map whose periodic points are counted by A061694.

Original entry on oeis.org

0, 0, 12, 216, 3500, 58494, 1028167, 18954072, 363991752, 7231521650, 147777013109, 3091874792274, 65993049570175, 1432803420182428, 31570847522072400, 704668366087255200, 15907964778448807820

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old name was: A061694 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n under that map.

			b(1)=0, b(3)=36 so a(3)=12.

Vaclav Kotesovec, Table of n, a(n) for n = 1..200
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Thomas Ward, Exactly realizable sequences. [local copy].

Cf. A061694.

Table[Sum[MoebiusMu[d] * Sum[Sum[((n/d)!/(i!*j!*(n/d - i - j)!))^3/6, {i, 1, n/d - j - 1}], {j, 1, n/d}], {d, Divisors[n]}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Sep 05 2019 *)

If b(n) is the n-th term of A061694, then a(n) = (1/n)*Sum_{d|n}mu(d)b(n/d).

a(n) ~ 3^(3*n + 1) / (8 * Pi^2 * n^3). - Vaclav Kotesovec, Sep 05 2019

Name clarified by Michel Marcus, May 14 2015

A091268 Number of orbits of length n under the map whose periodic points are counted by A061685.

Original entry on oeis.org

1, 4, 99, 6272, 876725, 232419936, 105471170140, 76095730062464, 82555139387847312, 128928209221144677400, 279860608037771819829980, 820360089598849358326307904, 3169977309466844379463315722484

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old name was: A061685 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n for that map.

			b(1)=1, b(2)=9, b(3)=298. Hence a(3)=(1/3)(b(3)-b(1))=99.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
Thomas Ward, Exactly realizable sequences. [local copy].

Cf. A061685.

If b(n) is the (n+1)th term of A061685, then a(n) = (1/n)*Sum_{d|n}mu(d)b(n/d).

Name clarified by Michel Marcus, May 14 2015

cp's OEIS Frontend

User: Thomas Ward

A341991 Multiplicative defect in a natural approximation for the terms of A341617.

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A341617 Repair factors for Stirling numbers of the second kind.

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A159282 Denominator of the rational coefficient in the main term in the dynamical analog of Mertens's theorem for a full n-dimensional shift, n >= 2.

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A159283 Numerator of the rational coefficient in the main term in the dynamical analog of Mertens's theorem for a full n-dimensional shift, n >= 12 (it is 1 for 2 <= n <= 11).

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A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4*sin^2(Pi*j*x) dx.

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A091112 Number of orbits of length n under the map whose periodic points are counted by A061686.

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Maple

Mathematica

PARI

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A091160 Number of orbits of length n under the map whose periodic points are counted by A061687.

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A133871 a(n) = the definite integral Integral_{0..1} Product_{j=1..n} 4sin^2(Pij*x) dx.