A006519 -id:A006519 - cp's OEIS frontend

A100863 Decimal expansion of the square of the constant (A100338) which has the continued fraction expansion equal to A006519 (highest power of 2 dividing n).

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

Offset: 1

Paul D. Hanna, Nov 20 2004

The continued fraction of this constant (A100864) has large partial quotients (A100865) that appear to be doubly exponential.

			1.83296703239600305442721954421041732405771656322721689779838977855718799...

Cf. A006519, A100338, A100864, A100865.

{CFM=contfracpnqn(vector(1500,n,2^valuation(n,2))); x=(CFM[1,1]/CFM[2,1])^2*1.0}

A100864 Continued fraction expansion of the square of the constant (A100338) which has the continued fraction equal to A006519 (highest power of 2 dividing n).

Original entry on oeis.org

1, 1, 4, 1, 74, 1, 8457, 1, 186282390, 1, 1, 1, 2, 1, 430917181166219, 11, 37, 1, 4, 2, 41151315877490090952542206046, 11, 5, 3, 12, 2, 34, 2, 9, 8, 1, 1, 2, 7, 13991468824374967392702752173757116934238293984253807017, 3, 4, 1, 3, 100, 4

Offset: 1

Paul D. Hanna, Nov 21 2004

Decimal expansion is 1.832967032396... (see A100863). Records are doubly exponential and form A100865.

Dzmitry Badziahin and Jeffrey Shallit, An Unusual Continued Fraction, arXiv:1505.00667 [math.NT], 2015.
Dzmitry Badziahin and Jeffrey Shallit, An unusual continued fraction, Proc. Amer. Math. Soc. 144 (2016), 1887-1896.

Cf. A006519, A100338, A100863, A100865.

{CFM=contfracpnqn(vector(650,n,2^valuation(n,2))); contfrac((CFM[1,1]/CFM[2,1])^2,71)}

A101120 Records in A101119, which forms the nonzero differences of A006519 and A003484.

Original entry on oeis.org

7, 22, 52, 112, 239, 494, 1004, 2024, 4071, 8166, 16356, 32736, 65503, 131038, 262108, 524248, 1048535, 2097110, 4194260, 8388560, 16777167, 33554382, 67108812, 134217672, 268435399, 536870854, 1073741764, 2147483584, 4294967231, 8589934526, 17179869116, 34359738296

Offset: 1

Simon Plouffe and Paul D. Hanna, Dec 02 2004

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,0,1,-3,2).

Cf. A003484, A006519, A101119, A101121, A101122.

LinearRecurrence[{3,-2,0,1,-3,2},{7,22,52,112,239,494},30] (* Harvey P. Dale, Jan 23 2023 *)

PARI
```
a(n)=2^(n+3)-2^((n-1)%4)-8*((n+3)\4)
```

def A101120(n): return (1<<(n+3))-(1<<((n-1)&3))-(((n+3)&-4)<<1) # Chai Wah Wu, Jul 10 2022

a(n) = A101119(2^(n-1)) for n>=1.
a(n) = 2^(n+3) - 2^((n-1)(mod 4)) - 8*floor((n+3)/4).
a(n) = 2^(n+3) - A003485(n+3). - Johannes W. Meijer, Oct 31 2012
From Chai Wah Wu, Apr 15 2017: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-4) - 3*a(n-5) + 2*a(n-6) for n > 6.
G.f.: x*(-x - 7)/((x - 1)^2*(x + 1)*(2*x - 1)*(x^2 + 1)). (End)
E.g.f.: (exp(x)*(32*exp(x) - 8*x - 27) - 4*cos(x) - cosh(x) - 2*sin(x) + sinh(x))/4. - Stefano Spezia, Jun 06 2023

A161800 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 2A006519(n) q^n/n ).

Original entry on oeis.org

1, 2, 0, 0, -6, -16, 0, 0, -8, 18, 0, 0, 112, 176, 0, 0, -86, -544, 0, 0, -752, -160, 0, 0, 1360, 2834, 0, 0, 1216, -5104, 0, 0, -5384, 3232, 0, 0, 10762, 18032, 0, 0, -8176, -68992, 0, 0, -59888, 48400, 0, 0, 130160, 143074, 0, 0, 47696, -343088, 0, 0

Offset: 0

Paul D. Hanna, Jul 19 2009

sign

A002129 forms the l.g.f. of log[ Sum_{n>=0} q^(n(n+1)/2) ], while
2*A006519 forms the l.g.f. of binary partitions (A000123) and
A006519(n) is the highest power of 2 dividing n.

			G.f.: A(q) = 1 + 2*q - 6*q^4 - 16*q^5 - 8*q^8 + 18*q^9 + 112*q^12 + 176*q^13 +...
log(A(q)) = 2*q - 4*q^2/2 + 8*q^3/3 - 40*q^4/4 + 12*q^5/5 - 16*q^6/6 +...
Sum_{n>=1} A002129(n)*q^n/n = log(1 + q + q^3 + q^6 + q^10 + q^15 +...),
Sum_{n>=1} 2*A006519(n)*x^n/n = log of the g.f. of binary partitions A000123.
QUADRASECTIONS:
Q_0(q) = 1 - 6*q - 8*q^2 + 112*q^3 - 86*q^4 - 752*q^5 + 1360*q^6 +...
Q_1(q) = 2 - 16*q + 18*q^2 + 176*q^3 - 544*q^4 - 160*q^5 + 2834*q^6 +...
The ratio Q_1(q)/Q_0(q) yields:
2 - 4*q + 10*q^2 - 20*q^3 + 36*q^4 - 64*q^5 + 110*q^6 - 180*q^7 +...
which appears to equal the g.f. of A127392.

Cf. A127392, quadrasections: A161801, A161802.

Cf. A002129, A006519, A000123.

{a(n)=local(L=sum(m=1, n,2*2^valuation(m,2)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

a(n) = 0 when n == 2 or 3 (mod 4).
Define the nonzero series QUADRASECTIONS:
Q_0(q) = Sum_{n>=0} a(4n)*q^n,
Q_1(q) = Sum_{n>=0} a(4n+1)*q^n, then:
Q_1(q)/Q_0(q) = series expansion of the elliptic function sqrt(k)/q^(1/4), where sqrt(k) = theta_2/theta_3, as described by A127392.
[The above statements are conjectures needing proof.]

A162581 G.f.: A(x) = exp( 2Sum_{n>=1} A006519(n)^2 x^n/n ), where A006519(n) = highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 6, 10, 26, 42, 86, 130, 258, 386, 694, 1002, 1754, 2506, 4134, 5762, 9346, 12930, 20198, 27466, 42330, 57194, 85750, 114306, 169602, 224898, 326934, 428970, 618138, 807306, 1144390, 1481474, 2084610, 2687746, 3732422, 4777098, 6591386

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 26*x^4 + 42*x^5 + 86*x^6 + ...
log(A(x))/2 = 2^0*x + 2^2*x^2 + 2^0*x^3/3 + 2^4*x^4/4 + 2^0*x^5/5 + 2^2*x^6/6 + 2^0*x^7/7 + 2^6*x^8/8 + ... + A006519(n)^2*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A162580, A162582, A006519, A000123.

nmax = 200; a[n_]:= SeriesCoefficient[Series[Exp[ Sum[2^(2*IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q,0,nmax}], n]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 04 2018 *)

{a(n)=local(L=sum(m=1,n,2*(2^valuation(m,2))^2*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}

A162584 G.f.: A(x) = exp( 2Sum_{n>=1} sigma(n)A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 8, 16, 50, 96, 240, 448, 1024, 1858, 3888, 6896, 13696, 23776, 44960, 76608, 139970, 234432, 414904, 684336, 1181568, 1921472, 3242928, 5206208, 8623104, 13679490, 22268752, 34941120, 56039936, 87036576, 137686048, 211822976

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

Log of the g.f. A(x) is formed from the term-wise product of the log of the g.f.s of the partition numbers A000041 and the binary partitions A000123.

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 16*x^3 + 50*x^4 + 96*x^5 + 240*x^6 + ...
log(A(x))/2 = x + 6*x^2/2 + 4*x^3/3 + 28*x^4/4 + 6*x^5/5 + 24*x^6/6 + 8*x^7/7 + 120*x^8/8 + ... + sigma(n)*A006519(n)*x^n/n + ...
The log of the g.f. of the Partition numbers (A000041) is:
x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 6*x^5/5 + 12*x^6/6 + ... + sigma(n)*x^n/n + ...
The log of the g.f. of the binary partitions (A000123) is:
x + x^2/2 + x^3/3 + 4*x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 + ... + A006519(n)*x^n/n + ...
From _Paul D. Hanna_, Jul 26 2009: (Start)
BISECTIONS begin:
B_0(q) = 1 + 8*q^2 + 50*q^4 + 240*q^6 + 1024*q^8 + 3888*q^10 + ...
B_1(q) = 2*q + 16*q^3 + 96*q^5 + 448*q^7 + 1858*q^9 + 6896*q^11 + ...
where 2*B_0(q)/B_1(q) = T16B(q):
T16B = 1/q + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + ...
which is a g.f. of A029839. (End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Cf. A000203, A006519, A000041, A000123.

Cf. A163228 (B_0), A163229 (B_1), A029839 (T16B); variant: A163129. - Paul D. Hanna, Jul 26 2009

eta[q_]:= q^(1/24)*QPochhammer[q]; nmax = 250; a[n_]:=SeriesCoefficient[ Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], n]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jul 03 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*2^(IntegerExponent[k, 2] + 1)*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 20 2020 *)
nmax = 40; CoefficientList[Series[Product[1/EllipticTheta[4, 0, x^(2^k)]^(2^k), {k, 0, 1 + Log[2, nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)

{a(n)=local(L=sum(m=1,n,2*sigma(m)*2^valuation(m,2)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}

From Paul D. Hanna, Jul 26 2009: (Start)
Define series BISECTIONS A(q) = B_0(q) + B_1(q), then
2*B_0(q)/B_1(q) = T16B(q) = q*eta(q^8)^6/(eta(q^4)^2*eta(q^16)^4), the McKay-Thompson series of class 16B for the Monster group (A029839). (End)
G.f.: 1/Product_{n>=0} Theta4(q^(2^n))^(2^n) = 1 / ( E(1)^2*E(2)^3*E(4)^6*E(8)^12* ... * E(2^n)^A042950(n) * ... ) where E(n) = Product_{k>=1} (1-q^(n*k)). - Joerg Arndt, Mar 20 2010
Compare to the previous formula: 1/Product_{n>=0} Theta3(q^(2^n))^(2^n) = Theta4(q). - Joerg Arndt, Aug 03 2011

A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Aug 26 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.

The equivalent sequence for partitions is A220482.

			----------------------------------------------------------
.             Diagram                Triangle
Compositions    of            of compositions (rows)
of 5          regions          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5           |_        |                                 5
1+4         |_|_      |                               1 4
2+3         |_  |     |                             2   3
1+1+3       |_|_|_    |                           1 1   3
3+2         |_    |   |                         3       2
1+2+2       |_|_  |   |                       1 2       2
2+1+2       |_  | |   |                     2   1       2
1+1+1+2     |_|_|_|_  |                   1 1   1       2
4+1         |_      | |                 4               1
1+3+1       |_|_    | |               1 3               1
2+2+1       |_  |   | |             2   2               1
1+1+2+1     |_|_|_  | |           1 1   2               1
3+1+1       |_    | | |         3       1               1
1+2+1+1     |_|_  | | |       1 2       1               1
2+1+1+1     |_  | | | |     2   1       1               1
1+1+1+1+1   |_|_|_|_|_|   1 1   1       1               1
.
Written as an irregular triangle in which row n lists the parts of the n-th region the sequence begins:
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,2,2,3,4;
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the regions of the last section of the set of compositions of r:
[1];
[1,2];
[1],[1,1,2,3];
[1],[1,2],[1],[1,1,1,1,2,2,3,4];
[1],[1,2],[1],[1,1,2,3],[1],[1,2],[1],[1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5];

Michael De Vlieger, Table of n, a(n) for n = 1..13312 (rows 1 <= n <= 2^11 = 2048).

Main triangle: Right border gives A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A090996, A186114, A187816, A187818, A206437, A220482, A228347, A228348, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

Table[Map[Length@ TakeWhile[IntegerDigits[#, 2], # == 1 &] &, Range[2^(# - 1), 2^# - 1]] &@ IntegerExponent[2 n, 2], {n, 32}] // Flatten (* Michael De Vlieger, May 23 2017 *)

A059991 a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))).

Original entry on oeis.org

1, 1, 4, 1, 16, 16, 64, 1, 256, 256, 1024, 256, 4096, 4096, 16384, 1, 65536, 65536, 262144, 65536, 1048576, 1048576, 4194304, 65536, 16777216, 16777216, 67108864, 16777216, 268435456, 268435456, 1073741824, 1, 4294967296, 4294967296

Offset: 1

Thomas Ward, Mar 08 2001

Number of points of period n in the simplest nontrivial disconnected S-integer dynamical system.

This sequence comes from the simplest disconnected S-integer system that is not hyperbolic. In the terminology of the papers referred to, it is constructed by choosing the under- lying field to be F_2(t), the element to be t and the nontrivial valuation to correspond to the polynomial 1+t. Since it counts periodic points, it satisfies the nontrivial congruence sum_{d|n}mu(d)a(n/d) = 0 mod n for all n and since it comes from a group automorphism it is a divisibility sequence.

			a(24) = 2^16 = 65536 because ord_2(24)=3, so 24-2^ord_2(24)=16.

R. Brown and J. L. Merzel, The number of Ducci sequences with a given period, Fib. Quart., 45 (2007), 115-121.
Vijay Chothi, Periodic Points in S-integer dynamical systems, PhD thesis, University of East Anglia, 1996.
Vijay Chothi, Graham Everest, Thomas Ward, S-integer dynamical systems: periodic points, J. Reine Angew Math. 489 (1997), 99-132.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Almost all S-integer dynamical systems have many periodic points, Ergodic Th. Dynam. Sys. 18 (1998), 471-486.
Index to divisibility sequences

Cf. A000079, A006519, A129760.

readlib(ifactors): for n from 1 to 100 do if n mod 2 = 1 then ord2 := 0 else ord2 := ifactors(n)[2][1][2] fi: printf(`%d,`, 2^(n-2^ord2)) od:

ord2[n_?OddQ] = 0; ord2[n_?EvenQ] := FactorInteger[n][[1, 2]]; a[n_] := 2^(n-2^ord2[n]); a /@ Range[34]
(* Jean-François Alcover, May 19 2011, after Maple prog. *)

More terms from James Sellers, Mar 15 2001

A162582 G.f.: A(x) = exp( 2Sum_{n>=1} A006519(n)^n x^n/n ), where A006519(n) = highest power of 2 dividing n.

Original entry on oeis.org

1, 2, 6, 10, 146, 282, 826, 1370, 4204986, 8408602, 25223066, 42037530, 615687706, 1189337882, 3483938586, 5778539290, 2305851850537847066, 4611703695297154842, 13835111074334385946, 23058518453371617050

Offset: 0

Paul D. Hanna, Jul 06 2009

nonn

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 146*x^4 + 282*x^5 + 826*x^6 + ...
log(A(x))/2 = 2^0*x + 2^2*x^2 + 2^0*x^3/3 + 2^8*x^4/4 + 2^0*x^5/5 + 2^6*x^6/6 + 2^0*x^7/7 + 2^24*x^8/8 + ... + A006519(n)^n*x^n/n + ...

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A162580, A162581, A155200, A006519, A000123.

nmax = 200; a[n_]:= SeriesCoefficient[Series[Exp[ Sum[2^(k*IntegerExponent[k, 2] + 1)*q^k/k, {k, 1, nmax}]], {q,0,nmax}], n]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jul 04 2018 *)

{a(n)=local(L=sum(m=1,n,2*(2^valuation(m,2))^m*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}

A193036 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^A006519(n), where A006519(n) is the highest power of 2 dividing n.

Original entry on oeis.org

1, 1, 1, 3, 10, 34, 112, 382, 1352, 4884, 17856, 66022, 246764, 930878, 3538788, 13542716, 52133416, 201746212, 784378792, 3062431132, 12001867312, 47197716460, 186187480816, 736582735738, 2921679555340, 11617001425938, 46294191726972, 184866924629832

Offset: 0

Paul D. Hanna, Jul 14 2011

nonn

Compare the g.f. to a g.f. C(x) of the Catalan numbers: x = Sum_{n>=1} x^n*C(-x)^(2*n-1).

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 34*x^5 + 112*x^6 + ...
The g.f. satisfies:
x = x*A(-x) + x^2*A(-x)^2 + x^3*A(-x) + x^4*A(-x)^4 + x^5*A(-x) + x^6*A(-x)^2 + x^7*A(-x) + x^8*A(-x)^8 + x^9*A(-x) + ... + x^n * A(-x)^A006519(n) + ...
where A006519 begins: [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,...].
The g.f. also satisfies:
x = x*A(-x)/(1-x^2) + x^2*A(-x)^2/(1-x^4) + x^4*A(-x)^4/(1-x^8) + x^8*A(-x)^8/(1-x^16) + x^16*A(-x)^16/(1-x^32) + x^32*A(-x)^32/(1-x^64) + ...
Related table.
The table of coefficients in A(-x)^(2^n) / (1 - x^(2*2^n)) begins:
n=0: [1, -1, 2, -4, 12, -38, 124, -420, 1476, -5304,  ...];
n=1: [1, -2, 3, -8, 28, -96, 324, -1124, 4024, -14684,  ...];
n=2: [1, -4, 10, -28, 95, -344, 1244, -4512, 16616, -62072, ...];
n=3: [1, -8, 36, -136, 514, -2008, 7924, -31176, 122495, ...];
n=4: [1, -16, 136, -848, 4500, -22032, 103480, -473520, ...];
n=5: [1, -32, 528, -6048, 54632, -418720, 2855088, ...];
n=6: [1, -64, 2080, -45888, 775120, -10720576, 126777952, ...];
n=7: [1, -128, 8256, -358016, 11750304, -311550592, 6955997376, ...];
...
where x = Sum_{n>=0} x^(2^n) * A(-x)^(2^n) / (1 - x^(2*2^n)).

Paul D. Hanna, Table of n, a(n) for n = 0..720

Cf. A193037, A193038, A193039, A193040, A006519.

{a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=1,#A,(-x)^m*Ser(A)^(2^valuation(m,2))),#A));if(n<0,0,A[n+1])}

G.f. satisfies: x = Sum_{n>=0} x^(2^n) * A(-x)^(2^n) / (1 - x^(2*2^n)).

cp's OEIS Frontend

A100863 Decimal expansion of the square of the constant (A100338) which has the continued fraction expansion equal to A006519 (highest power of 2 dividing n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A100864 Continued fraction expansion of the square of the constant (A100338) which has the continued fraction equal to A006519 (highest power of 2 dividing n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A101120 Records in A101119, which forms the nonzero differences of A006519 and A003484.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A161800 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 2*A006519(n) * q^n/n ).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A162581 G.f.: A(x) = exp( 2*Sum_{n>=1} A006519(n)^2 * x^n/n ), where A006519(n) = highest power of 2 dividing n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A162584 G.f.: A(x) = exp( 2*Sum_{n>=1} sigma(n)*A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A059991 a(n) = 2^(n-2^ord_2(n)) (or 2^(n-A006519(n))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A161800 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 2A006519(n) q^n/n ).

A162581 G.f.: A(x) = exp( 2Sum_{n>=1} A006519(n)^2 x^n/n ), where A006519(n) = highest power of 2 dividing n.

A162584 G.f.: A(x) = exp( 2Sum_{n>=1} sigma(n)A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

A162582 G.f.: A(x) = exp( 2Sum_{n>=1} A006519(n)^n x^n/n ), where A006519(n) = highest power of 2 dividing n.