A048651 -id:A048651 - cp's OEIS frontend

A005283 Number of permutations of (1,...,n) having n-5 inversions (n>=5).

1, 5, 20, 76, 285, 1068, 4015, 15159, 57486, 218895, 836604, 3208036, 12337630, 47572239, 183856635, 712033264, 2762629983, 10736569602, 41788665040, 162869776650, 635562468075, 2482933033659, 9710010151831, 38008957336974, 148912655255315, 583885852950802

Offset: 5

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(6)=5 because we have 213456, 132456, 124356, 123546, and 123465.

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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 5..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.

Cf. A008302, A000707, A001892, A001893, A001894, A005284, A005285, A048651.

f := (x,n)->product((1-x^j)/(1-x),j=1..n); seq(coeff(series(f(x,n),x,n+2),x,n-5), n=5..40); # Barbara Haas Margolius, May 31 2001

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-5}],{n,5,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-6)/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

A005284 Number of permutations of (1,...,n) having n-6 inversions (n>=6).

Original entry on oeis.org

1, 6, 27, 111, 440, 1717, 6655, 25728, 99412, 384320, 1487262, 5762643, 22357907, 86859412, 337879565, 1315952428, 5131231668, 20029728894, 78265410550, 306109412100, 1198306570554, 4694809541046, 18407850118383

Offset: 6

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(7)=6 because we have 2134567, 1324567, 1243567, 1235467, 1234657 and 1234576.

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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 6..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.

Cf. A008302, A000707, A001892, A001893, A001894, A005283, A005285, A048651.

g := proc(n,k) option remember; if k=0 then return(1) else if (n=1 and k=1) then return(0) else if (k<0 or k>binomial(n,2)) then return(0) else g(n-1,k)+g(n,k-1)-g(n-1,k-n) end if end if end if end proc; seq(g(j+6,j),j=0..30); # Barbara Haas Margolius, May 31 2001

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-6}],{n,6,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-7)/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

A005285 Number of permutations of (1,...,n) having n-7 inversions (n>=7).

Original entry on oeis.org

1, 7, 35, 155, 649, 2640, 10569, 41926, 165425, 650658, 2554607, 10020277, 39287173, 154022930, 603919164, 2368601685, 9293159292, 36476745510, 143239635450, 562744102479, 2211876507387, 8697839966552, 34218338900591

Offset: 7

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(8)=7 because we have 21345678, 13245678, 12435678, 12354678, 12346578, 12345768, and 12345687.

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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 7..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.

Cf. A008302, A000707, A001892, A001893, A001894, A005283, A005284, A048651.

g := proc(n,k) option remember; if k=0 then return(1) else if (n=1 and k=1) then return(0) else if (k<0 or k>binomial(n,2)) then return(0) else g(n-1,k)+g(n,k-1)-g(n-1,k-n) end if end if end if end proc; seq(g(j+7,j),j=0..30); # Barbara Haas Margolius, May 31 2001

Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-7}],{n,7,25}] (* Vaclav Kotesovec, Mar 16 2014 *)

a(n) = 2^(2*n-8)/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

A101872 Number of Abelian groups of order 2n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 5, 2, 2, 1, 3, 1, 2, 1, 7, 1, 4, 1, 3, 1, 2, 1, 5, 2, 2, 3, 3, 1, 2, 1, 11, 1, 2, 1, 6, 1, 2, 1, 5, 1, 2, 1, 3, 2, 2, 1, 7, 2, 4, 1, 3, 1, 6, 1, 5, 1, 2, 1, 3, 1, 2, 2, 15, 1, 2, 1, 3, 1, 2, 1, 10, 1, 2, 2, 3, 1, 2, 1, 7, 5, 2, 1, 3, 1, 2, 1, 5, 1, 4, 1, 3, 1, 2, 1, 11, 1, 4, 2, 6, 1, 2, 1, 5

Offset: 1

N. J. A. Sloane, Jan 28 2005

Antti Karttunen, Table of n, a(n) for n = 1..65537

Bisection of A000688.
Cf. also A101876 (bisection of this sequence).
Cf. A000041, A021002, A048651.

Table[FiniteAbelianGroupCount[2 k], {k, 1, 100}] (* Geoffrey Critzer, Dec 29 2014 *)

A101872(n) = factorback(apply(e -> numbpart(e),factor(2*n)[,2])); \\ Antti Karttunen, Sep 27 2018

a(n) = A000688(2n).
Multiplicative with a(2^k) = A000041(1+k), and for odd primes p, a(p^k) = A000041(k), where A000041(k) is the number of partitions of k. - Antti Karttunen, Sep 27 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * (1-A048651) * A021002 = 3.26425865613408900779... . - Amiram Eldar, Sep 23 2023

More terms from Joshua Zucker, May 10 2006

A101876 Number of Abelian groups of order 4n.

Original entry on oeis.org

2, 3, 2, 5, 2, 3, 2, 7, 4, 3, 2, 5, 2, 3, 2, 11, 2, 6, 2, 5, 2, 3, 2, 7, 4, 3, 6, 5, 2, 3, 2, 15, 2, 3, 2, 10, 2, 3, 2, 7, 2, 3, 2, 5, 4, 3, 2, 11, 4, 6, 2, 5, 2, 9, 2, 7, 2, 3, 2, 5, 2, 3, 4, 22, 2, 3, 2, 5, 2, 3, 2, 14, 2, 3, 4, 5, 2, 3, 2, 11, 10, 3, 2, 5, 2, 3, 2, 7, 2, 6, 2, 5, 2, 3, 2, 15, 2, 6, 4, 10, 2

Offset: 1

N. J. A. Sloane, Jan 28 2005

Antti Karttunen, Table of n, a(n) for n = 1..65537

Bisection of A101872, quadrisection of A000688.

Cf. A000041, A021002, A048651.

a[n_] := FiniteAbelianGroupCount[4*n]; Array[a, 100] (* Amiram Eldar, Sep 23 2023*)

A101876(n) = factorback(apply(e -> numbpart(e),factor(4*n)[,2])); \\ Antti Karttunen, Sep 27 2018

a(n) = A000688(4*n). - Antti Karttunen, Sep 27 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (4 - 6 * A048651) * A021002 = 5.20306278505563943501... . - Amiram Eldar, Sep 23 2023

More terms from Joshua Zucker, May 10 2006

A104488 Number of Hamiltonian groups of order n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

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

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.

T. D. Noe, 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.
Tomaž Pisanski and Thomas W. Tucker, The genus of low rank hamiltonian groups, Discrete Math. 78 (1989), 157-167.
Eric Weisstein's World of Mathematics, Hamiltonian Group.

Cf. A000265, A000688, A021002, A048651, A104404, A104404, A104452, A104453.

orders[n_]:=Map[Last, FactorInteger[n]]; a[n_]:=Apply[Times, Map[PartitionsP, orders[n]]]; e[n_]:=n/ 2^IntegerExponent[n, 2]; h[n_]/;Mod[n, 8]==0:=a[e[n]]; h[n_]:=0;
(* Second program: *)
a[n_] := If[Mod[n, 8]==0, FiniteAbelianGroupCount[n/2^IntegerExponent[n, 2]], 0]; Array[a, 102] (* Jean-François Alcover, Sep 14 2019 *)

a(n)={my(e=valuation(n, 2)); if(e<3, 0, my(f=factor(n/2^e)[, 2]); prod(i=1, #f, numbpart(f[i])))} \\ Andrew Howroyd, Aug 08 2018

Let n = 2^e*o, where e = e(n) >= 0 and o = o(n) is an odd number. The number h(n) of Hamiltonian groups of order n is given by h(n) = 0, if e(n) < 3 and h(n) = a(o(n)), otherwise, where a(n) = A000688(n) denotes the number of Abelian groups of order n.
a(8*n) = A000688(A000265(n)), a(n) = 0 for n mod 8 <> 0. - Andrew Howroyd, Aug 08 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A021002 * A048651 / 4 = 0.16568181590156732257... . - Amiram Eldar, Sep 23 2023

A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 45, 64, 68, 72, 76, 100, 105, 110, 115, 144, 150, 156, 162, 196, 203, 210, 217, 512, 528, 544, 560, 648, 666, 684, 702, 800, 820, 840, 860, 968, 990, 1012, 1034, 1728, 1764, 1800, 1836, 2028, 2067, 2106, 2145, 2352

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-4 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(26)=floor(26/4^0)*floor(26/4^1)*floor(26/4^2)=26*6*1=156; a(34)=544 since 34=202(base-4) and so
a(34)=202*20*2(base-4)=34*8*2=544.

Cf. A048651, A132020, A100221, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Recurrence: a(n)=n*a(floor(n/4)); a(n*4^m)=n^m*4^(m(m+1)/2)*a(n).

a(k*4^m)=k^(m+1)*4^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_4(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_4(n)))/4^((1+floor(log_4(n)))*floor(log_4(n))/2); equality holds for n=k*4^m, 0=0. b(n) can also be written n^(1+floor(log_4(n)))/4^A000217(floor(log_4(n))).

Also: a(n)<=2^(1/4)*n^((1+log_4(n))/2)=1.189207...*4^A000217(log_4(n)), equality holds for n=2*4^m, m>=0.

a(n)>c*b(n), where c=0.4194224417951075977... (see constant A132020).

Also: a(n)>c*2^(1/4)*n^((1+log_4(n))/2)=0.498780...*4^A000217(log_4(n)).

lim inf a(n)/b(n)=0.4194224417951075977..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_4(n))/2)=0.4194224417951075977...*2^(1/4), for n-->oo.

lim sup a(n)/n^((1+log_4(n))/2)=2^(1/4), for n-->oo.

lim inf a(n)/a(n+1)=0.4194224417951075977... for n-->oo (see constant A132020).

A152537 Convolution sequence: this sequence convolved with A000041 gives powers of 2, (A000079).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 18, 37, 74, 148, 296, 592, 1183, 2366, 4732, 9463, 18926, 37852, 75704, 151408, 302816, 605632, 1211265, 2422530, 4845060, 9690120, 19380241, 38760482, 77520964, 155041928, 310083856, 620167712, 1240335424, 2480670848, 4961341696, 9922683391

Offset: 0

Gary W. Adamson, Dec 06 2008

nonn

Terms are very similar to those of A178841. - Georg Fischer, Mar 23 2019

			a(5) = 9 = 32 - 23 = (32 - ((7,5,3,2,1) dot (1,1,1,2,4)))
(1,1,2,3) convolved with (1,1,1,2) = 8, where (1,1,2,3...) = the first four partition numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A010815, A000041, A000079, A152538, A178841.

a:= proc(n) option remember;
      2^n-add(combinat[numbpart](j)*a(n-j), j=1..n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2025

nmax = 40; CoefficientList[Series[Product[1-x^k, {k, 1, nmax}] / (1-2*x), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 02 2018 *)

/* computation by definition (division of power series) */
N=55;
A000079=vector(N,n,2^(n-1));
S000079=Ser(A000079);
A000041=vector(N,n,numbpart(n-1));
S000041=Ser(A000041);
S152537=S000079/S000041;
A152537=Vec(S152537) /* show terms */  /* Joerg Arndt, Feb 06 2011 */

/* computation using power series eta(x) and 1/(1-2*x) */
x='x+O('x^55);  S152537=eta(x)/(1-2*x);
A152537=Vec(S152537) /* show terms */  /* Joerg Arndt, Feb 06 2011 */

Construct an array of rows such that n-th row = partial sums of (n-1)-th row of A010815: (1, -1, -1, 0, 0, 1, 0, 1,...).
A152537 = sums of antidiagonal terms of the array.
The sequence may be obtained directly from the following set of operations:
Our given sequence = A000041: (1, 1, 2, 3, 5, 7, 11,...). Delete the first "1" then consider (1, 2, 3, 5, 7, 11,...) as an operator Q which we write in reverse with 1,2,3,...terms for each operation. Letting R = the target sequence (1,2,4,8,...); we begin a(0) = 1, a(1) = 1, then perform successive operations of: "next term in (1,2,4,...) - dot product of Q*R" where Q is written right to left and R (the ongoing result) written left to right).
Examples: Given 4 terms Q, R, we have: (5,3,2,1) dot (1,1,1,2) = (5+3+2+2) = 12, which we subtract from 16, = 4.
Given 5 terms of Q,R and A152537, we have (7,5,3,2,1) dot (1,1,1,2,4) = 23 which is subtracted from 32 giving 9. Continue with analogous operations to generate the series.
a(n) = Sum_{j=0..n} A010815(j)*2^(n-j). G.f.: A000079(x)/A000041(x) = A010815(x)/(1-2x), where A......(x) denotes the g.f. of the associated sequence. - R. J. Mathar, Dec 09 2008
a(n) ~ c * 2^n, where c = A048651 = 0.28878809508660242127889972192923078... - Vaclav Kotesovec, Jun 02 2018
a(n) = 2^n - Sum_{j=1..n} A000041(j)*a(n-j). - Alois P. Heinz, Feb 02 2025

A259401 a(n) = Sum_{k=0..n} 2^(n-k)*p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 3, 8, 19, 43, 93, 197, 409, 840, 1710, 3462, 6980, 14037, 28175, 56485, 113146, 226523, 453343, 907071, 1814632, 3629891, 7260574, 14522150, 29045555, 58092685, 116187328, 232377092, 464757194, 929518106, 1859040777, 3718087158, 7436181158, 14872370665

Offset: 0

Vaclav Kotesovec, Jun 26 2015

nonn

In general, Sum_{k=0..n} (m^(n-k) * p(k)) ~ m^n / QPochhammer[1/m, 1/m], for m > 1.

Alois P. Heinz, Table of n, a(n) for n = 0..3320

Cf. A000041, A048651, A090764, A259400, A292746.

a:= proc(n) option remember; `if`(n<0, 0,
      2*a(n-1)+combinat[numbpart](n))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 03 2019

Table[Sum[2^(n-k)*PartitionsP[k],{k,0,n}],{n,0,50}]

a(n) = sum(k=0, n, 2^(n-k)*numbpart(k)); \\ Michel Marcus, Dec 03 2019

a(n) ~ c * 2^n, where c = 1/A048651 = 1/QPochhammer[1/2, 1/2] = 3.462746619455...

G.f.: (1/(1 - 2*x)) * Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Dec 03 2019

A335764 Decimal expansion of Sum_{k>=1} sigma(k)/(k*2^k) where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 21 2020

			1.242062094812414945797845481894629668973403978250425...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Lambert Series.
Wikipedia, Lambert series.

Cf. A000203, A017665, A017666, A048651, A065442, A066524, A066766, A256936, A335763.

RealDigits[Sum[1/n/(2^n - 1), {n, 1, 500}], 10, 100][[1]]
RealDigits[-Log[QPochhammer[1/2]], 10, 120][[1]] (* Vaclav Kotesovec, Feb 18 2021 *)

suminf(x = 1, sigma(x)/(x*2^x)) \\ David A. Corneth, Jun 21 2020

Equals Sum_{k>=1} (A017665(k)/A017666(k))/2^k.
Equals Sum_{k>=1} 1/(k*(2^k - 1)) = Sum_{k>=1} 1/A066524(k).
Equals -Sum_{k>=1} log(1-2^(-k)).
Equals -log(A048651). - Amiram Eldar, Feb 19 2022

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A005283 Number of permutations of (1,...,n) having n-5 inversions (n>=5).

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A005284 Number of permutations of (1,...,n) having n-6 inversions (n>=6).

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A005285 Number of permutations of (1,...,n) having n-7 inversions (n>=7).

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A101872 Number of Abelian groups of order 2n.

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A101876 Number of Abelian groups of order 4n.

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A104488 Number of Hamiltonian groups of order n.

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A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

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