A048651 -id:A048651 - cp's OEIS frontend

A128566 Number of permutations of {1..n} with n inversions.

1, 0, 0, 1, 5, 22, 90, 359, 1415, 5545, 21670, 84591, 330121, 1288587, 5032235, 19664205, 76893687, 300895513, 1178290263, 4617369760, 18106447251, 71048746505, 278966179936, 1095987764828, 4308300939450, 16944940572831, 66680029591816, 262519664110588

Offset: 0

Paul D. Hanna, Mar 12 2007

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1665

Diagonal of A008302 (Mahonian numbers).
Column 2 of A128564.
Cf. A128565 (column 1), A214086, A048651.

a:= n-> coeff(series(mul((1-q^j)/(1-q), j=1..n), q, n+1), q, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 05 2013

Table[SeriesCoefficient[QPochhammer[x, x, n]/(1-x)^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 13 2016 *)

{a(n)=polcoeff(prod(j=1, n, (1-q^j)/(1-q)),n,q)}

a(n) = A008302(n,n) = coefficient of q^n in the q-factorial of n.
a(n) = T(n,n) with T(n,k) = T(n-1,k) + Sum_{j=1..n-1} T(n-1,k-j) for n>=0, k>0; T(n,k) = 0 for n<0; T(n,0) = 1 for n>=0. - Alois P. Heinz, Mar 07 2013
a(n) ~ c * 2^(2*n-1) / sqrt(Pi*n), where c = A048651 = QPochhammer[1/2] = 0.28878809508660242127889972192923... . - Vaclav Kotesovec, Sep 07 2014

Edited by Alois P. Heinz, Mar 05 2013

A132267 Decimal expansion of Product_{k>0} (1-1/11^k).

Original entry on oeis.org

9, 0, 0, 8, 3, 2, 7, 0, 6, 8, 0, 9, 7, 1, 5, 2, 7, 9, 9, 4, 9, 8, 6, 2, 6, 9, 4, 7, 6, 0, 6, 4, 7, 7, 4, 4, 7, 6, 2, 4, 9, 1, 1, 9, 2, 2, 1, 6, 6, 3, 9, 5, 2, 4, 0, 2, 1, 4, 6, 1, 7, 2, 4, 8, 8, 0, 6, 5, 7, 0, 8, 7, 0, 6, 7, 0, 9, 7, 5, 8, 5, 6, 7, 0, 0, 1, 6, 3, 9, 2, 9, 9, 1, 9, 9, 2, 8, 3, 5, 6, 4, 6, 5, 2, 0

Offset: 0

Hieronymus Fischer, Aug 20 2007

			0.900832706809715279949862694760...

G. C. Greubel, Table of n, a(n) for n = 0..2000
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A000203, A027879, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323, A132324, A132325, A132326.

digits = 105; NProduct[1-1/11^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/11, 1/11], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(x=1, 1-1/11^x) \\ Altug Alkan, Dec 20 2015

Equals exp(-Sum_{n>0} sigma_1(n)/(n*11^n)) = exp(-Sum_{n>0} A000203(n)/(n*11^n)).
Equals (1/11; 1/11){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(11)) * exp(log(11)/24 - Pi^2/(6*log(11))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(11))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027879(n). (End)

A303346 Expansion of Product_{n>=1} ((1 + 2x^n)/(1 - 2x^n))^(1/2).

Original entry on oeis.org

1, 2, 4, 10, 18, 38, 72, 142, 260, 510, 940, 1814, 3362, 6490, 12112, 23466, 44114, 85766, 162516, 317190, 604806, 1184682, 2271248, 4461514, 8591784, 16916490, 32696708, 64496130, 125037142, 247007142, 480077432, 949510526, 1849375796, 3661330398, 7144215452

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A032302, A070933, A261584, A303307, A303345, A370709, A370713.

nmax = 30; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[Sqrt[-QPochhammer[-2, x] / (3*QPochhammer[2, x])], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+2*x^k)/(1-2*x^k))^(1/2)))

a(n) ~ 2^n / sqrt(c*Pi*n), where c = A048651 * A083864 = 1/2 * Product_{j>=1} (2^j-1)/(2^j+1) = 0.06056210400129025123042464659093375290492912341... - Vaclav Kotesovec, Apr 22 2018

A005327 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

Original entry on oeis.org

1, 0, 1, 6, 91, 2820, 177661, 22562946, 5753551231, 2940064679040, 3007686166657921, 6156733583148764286, 25211824022994189751171, 206510050572345408251841660, 3383254158526734823389921915781

Offset: 1

N. J. A. Sloane

nonn

q-Subfactorial for q=2. - Vladimir Reshetnikov, Sep 12 2016

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.
T. L. Greenough and T. Lockman, Representation and enumeration of interval orders and semiorders, Ph.D. Thesis, Dartmouth, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..50
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976. [Annotated scanned copy]
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Subfactorial, q-Factorial, q-Analog.

Cf. A002820, A005329, A005321.

p:=proc(n) if n=0 then 1 else product(2^i-1,i=1..n) fi end: a:=n->p(n-1)*sum((-1)^j/p(j),j=0..n-1): seq(a(n),n=1..17); # Emeric Deutsch, Jan 23 2005

a[1] = 1; a[n_] := a[n] = (2^(n-1)-1)*a[n-1] + (-1)^(n-1); Array[a, 15] (* Jean-François Alcover, Apr 05 2016, after Max Alekseyev *)
With[{q = 2}, Table[QFactorial[n, q] Sum[(-1)^k/QFactorial[k, q], {k, 0, n}], {n, 0, 20}]] (* Vladimir Reshetnikov, Sep 12 2016 *)

For n>1, a(n) = (2^(n-1)-1)*a(n-1) + (-1)^(n-1). - Max Alekseyev, Feb 23 2012
a(n) = p(n-1)*sum((-1)^j/p(j), j=0..n-1), where p(0) = 1, p(k) = product(2^i-1, i=1..k) for k>0. - Emeric Deutsch, Jan 23 2005
a(n) ~ A048651^2 * 2^(n*(n-1)/2). - Vaclav Kotesovec, Oct 09 2019

More terms from Max Alekseyev, Apr 27 2010

A007950 Binary sieve: delete every 2nd number, then every 4th, 8th, etc.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 17, 21, 25, 27, 29, 33, 35, 37, 43, 49, 51, 53, 57, 59, 65, 67, 69, 73, 75, 77, 81, 85, 89, 91, 97, 101, 107, 109, 113, 115, 117, 121, 123, 129, 131, 133, 137, 139, 145, 149, 153, 155, 157, 161, 163, 165, 171, 173, 177, 179, 181, 185, 187, 195, 197

Offset: 1

R. Muller

From Charles T. Le (charlestle(AT)yahoo.com), Mar 22 2004: (Start)
This sequence and A007951 are particular cases of the Smarandache n-ary sequence sieve (for n=2 and respectively n=3).
Definition of Smarandache n-ary sieve (n >= 2): Starting to count on the natural numbers set at any step from 1: - delete every n-th numbers; - delete, from the remaining numbers, every (n^2)-th numbers; ... and so on: delete, from the remaining ones, every (n^k)-th numbers, k = 1, 2, 3, ... .
Conjectures: there are infinitely many primes that belong to this sequence; also infinitely many composite numbers.
Smarandache general-sequence sieve: Let u_i > 1, for i = 1, 2, 3, ..., be a strictly increasing positive integer sequence. Then from the natural numbers: - keep one number among 1, 2, 3, ..., u_1 - 1 and delete every u_1 -th numbers; - keep one number among the next u_2 - 1 remaining numbers and delete every u_2 -th numbers; ... and so on, for step k (k >= 1): - keep one number among the next u_k - 1 remaining numbers and delete every u_k -th numbers; ... (End)
Certainly this sequence contains infinitely many composite numbers, as it has finite density A048651, while the primes have zero density. - Franklin T. Adams-Watters, Feb 25 2011

F. Smarandache, Properties of Numbers, 1972.

T. D. Noe, Table of n, a(n) for n = 1..10000
C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
F. Smarandache, Only Problems, Not Solutions!, 4th ed., 1993, Problem 95.
Index entries for sequences generated by sieves

Cf. A007951, A000959, A048651.

t = Range@200; f[n_] := Block[{k = 2^n}, t = Delete[t, Table[{k}, {k, k, Length@t, k}]]]; Do[ f@n, {n, 6}]; t (* Robert G. Wilson v, Sep 14 2006 *)

More terms from Robert G. Wilson v, Sep 14 2006

A001892 Number of permutations of (1,...,n) having n-2 inversions (n>=2).

Original entry on oeis.org

1, 2, 5, 15, 49, 169, 602, 2191, 8095, 30239, 113906, 431886, 1646177, 6301715, 24210652, 93299841, 360490592, 1396030396, 5417028610, 21056764914, 81978913225, 319610939055, 1247641114021, 4875896455975, 19075294462185, 74696636715792, 292758662041150

Offset: 2

N. J. A. Sloane, R. K. Guy

nonn

Sequence is a diagonal of the triangle A008302 (number of permutations of (1,...,n) with k inversions; see Table 1 of the Margolius reference). - Emeric Deutsch, Aug 02 2014

			a(4)=5  because we have 1342, 1423, 2143, 2314, and 3124.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.14., p.356.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 2..1000
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
E. Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 92-99 only.

Cf. A008302, A048651.

f := (x,n)->product((1-x^j)/(1-x),j=1..n); seq(coeff(series(f(x,n),x,n+2),x,n-2), n=2..40);

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-2}],{n,2,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-3)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.288788095... (see A048651). - corrected by Vaclav Kotesovec, Mar 16 2014

More terms, Maple code, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001

Definition clarified by Emeric Deutsch, Aug 02 2014

A001893 Number of permutations of (1,...,n) having n-3 inversions (n>=3).

Original entry on oeis.org

1, 3, 9, 29, 98, 343, 1230, 4489, 16599, 61997, 233389, 884170, 3366951, 12876702, 49424984, 190297064, 734644291, 2842707951, 11022366544, 42815701060, 166583279325, 649063995030, 2532267577126, 9891097066760, 38676401680776, 151381995733542, 593053313030007

Offset: 3

N. J. A. Sloane

nonn

Sequence is a diagonal of the triangle A008302 (number of permutations of (1,...,n) with k inversions; see Table 1 of the Margolius reference). - Emeric Deutsch, Aug 02 2014

			a(4)=3  because we have 1243, 1324, and 2134.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.14., p.356.
R. K. Guy, personal communication.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 15.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 3..1000
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
E. Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 92-99 only.

Cf. A008302, A048651.

f := (x,n)->product((1-x^j)/(1-x),j=1..n); seq(coeff(series(f(x,n),x,n+2),x,n-3), n=3..40); # Barbara Haas Margolius, May 31 2001

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-3}],{n,3,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-4)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.2887880951... (see A048651). - corrected by Vaclav Kotesovec, Mar 16 2014

More terms, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001

Definition clarified by Emeric Deutsch, Aug 02 2014

A001894 Number of permutations of {1,...,n} having n-4 inversions (n>=4).

Original entry on oeis.org

1, 4, 14, 49, 174, 628, 2298, 8504, 31758, 119483, 452284, 1720774, 6574987, 25214332, 96997223, 374153699, 1446677555, 5605337934, 21758936146, 84604366100, 329453055975, 1284626463105, 5015200610785, 19601107218591, 76685359017750, 300294650988857, 1176939165980809

Offset: 4

N. J. A. Sloane

nonn

Sequence is a diagonal of the triangle A008302 (number of permutations of (1,...,n) with k inversions; see Table 1 of the Margolius reference). - Emeric Deutsch, Aug 02 2014

			a(5)=4  because we have 21345, 13245, 12435, and 12354.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.14., p.356.
R. K. Guy, personal communication.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 4..1000
R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
E. Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 92-99 only.

Cf. A008302, A048651.

f := (x,n)->product((1-x^j)/(1-x),j=1..n); seq(coeff(series(f(x,n),x,n+2),x,n-4), n=4..40); # Barbara Haas Margolius, May 31 2001

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-4}],{n,4,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-5)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.2887880951... (see A048651). - corrected by Vaclav Kotesovec, Mar 16 2014

More terms, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001

Definition clarified by Emeric Deutsch, Aug 02 2014

A104404 Number of groups of order n all of whose subgroups are normal.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 2, 12, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 6, 5, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1

Offset: 1

Boris Horvat (Boris.Horvat(AT)fmf.uni-lj.si), Gasper Jaklic (Gasper.Jaklic(AT)fmf.uni-lj.si), Tomaz Pisanski, Apr 19 2005

A finite non-Abelian group has all of its subgroups normal precisely when it is the direct product of the quaternion group of order 8, a (possibly trivial) elementary Abelian 2-group, and an Abelian group of odd order. [Carmichael, p. 114] - Eric M. Schmidt, Jan 12 2014

Robert D. Carmichael, Introduction to the Theory of Groups of Finite Order, New York, Dover, 1956.
John C. Lennox and Stewart. E. Stonehewer, Subnormal Subgroups of Groups, Oxford University Press, 1987.

Hans Havermann, Table of n, a(n) for n = 1..10000
Boris Horvat, Gašper Jaklič, and Tomaž Pisanski, On the number of hamiltonian groups, Mathematical Communications, Vol. 10, No. 1 (2005), pp. 89-94; arXiv preprint, arXiv:math/0503183 [math.CO], 2005.
Eric Weisstein's World of Mathematics, Abelian Group.
Eric Weisstein's World of Mathematics, Hamiltonian Group.

Cf. A000001, A000688, A021002, A048651, A104488.

orders[n_]:=Map[Last, FactorInteger[n]]; b[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=b[e[n]]; h[n_]:=0; a[n_]:= b[n]+h[n];

a(n)={my(e=valuation(n, 2)); my(f=factor(n/2^e)[, 2]); prod(i=1, #f, numbpart(f[i]))*(numbpart(e) + (e>=3))} \\ Andrew Howroyd, Aug 08 2018

The number a(n) of all groups of order n all of whose subgroups are normal is given as a(n) = b(n) + h(n), where b(n) denotes the number of Abelian groups of order n and h(n) denotes the number of Hamiltonian groups of order n.
a(n) = A000688(n) + A104488(n). - Andrew Howroyd, Aug 08 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A021002 * (1 + A048651/4) = 2.46053840757488111675... . - Amiram Eldar, Sep 23 2023

Keyword:mult added by Andrew Howroyd, Aug 08 2018

A203011 (n-1)-st elementary symmetric function of {1,3,7,15,31,63,...-1+2^n}.

Original entry on oeis.org

1, 4, 31, 486, 15381, 978768, 124918731, 31932406170, 16337382642945, 16723323142761060, 34243057328337866295, 140246638967945496322350, 1148847521944847479468879725, 18822284044001939139425413111800, 616761496621711735518439444437389475

Offset: 1

Clark Kimberling, Dec 29 2011

nonn

Clark Kimberling, Table of n, a(n) for n = 1..50

Cf. A122743.

SymmPolyn := proc(L::list,n::integer)
    local c,a,sel;
    a :=0 ;
    sel := combinat[choose](nops(L),n) ;
    for c in sel do
        a := a+mul(L[e],e=c) ;
    end do:
    a;
end proc:
A203011 := proc(n)
    local L,k ;
    L := [seq(2^k-1,k=1..n)] ;
    SymmPolyn(L,n-1) ;
end proc: # R. J. Mathar, Sep 23 2016

f[k_] := -1 + 2^k; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A203011 *)
Table[Product[2^k-1,{k,1,n}] * Sum[1/(2^k-1),{k,1,n}],{n,1,16}] (* Vaclav Kotesovec, Sep 06 2014 *)

a(n) = c * 2^(n*(n+1)/2), where c = A048651 * A065442 = 0.4639944324508904477884... . - Vaclav Kotesovec, Oct 10 2016

cp's OEIS Frontend

A128566 Number of permutations of {1..n} with n inversions.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A132267 Decimal expansion of Product_{k>0} (1-1/11^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A303346 Expansion of Product_{n>=1} ((1 + 2*x^n)/(1 - 2*x^n))^(1/2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A005327 Certain subgraphs of a directed graph (inverse binomial transform of A005321).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A007950 Binary sieve: delete every 2nd number, then every 4th, 8th, etc.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Extensions

A001892 Number of permutations of (1,...,n) having n-2 inversions (n>=2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A001893 Number of permutations of (1,...,n) having n-3 inversions (n>=3).

Views

Author

Keywords

Comments

Examples

References

Links

A303346 Expansion of Product_{n>=1} ((1 + 2x^n)/(1 - 2x^n))^(1/2).