A000123 -id:A000123 - cp's OEIS frontend

A162588 G.f.: A(x) = exp( 2Sum_{n>=1} 2^n/A006519(n) x^n/n ), where A006519(n) = highest power of 2 dividing n.

1, 4, 10, 24, 52, 112, 240, 512, 1060, 2192, 4552, 9440, 19408, 39872, 81984, 168448, 342632, 696736, 1421200, 2897856, 5891872, 11976064, 24361856, 49543168, 100329952, 203147136, 411939264, 835168512, 1690383744, 3420860928

Offset: 0

Paul D. Hanna, Jul 07 2009

nonn

			G.f.: A(x) = 1 + 4*x + 10*x^2 + 24*x^3 + 52*x^4 + 112*x^5 + 240*x^6 + ...
log(A(x))/2 = 2*x + 2*x^2/2 + 8*x^3/3 + 4*x^4/4 + 32*x^5/5 + 32*x^6/6 + 128*x^7/7 + ...

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

Cf. A006519, A000123.

nmax = 150; a[n_]:= SeriesCoefficient[Series[Exp[Sum[2^(k + 1 - IntegerExponent[k, 2])*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=2*sum(m=1,n,2^(m-valuation(m,2))*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}

A178855 Partial sums of A033485.

Original entry on oeis.org

1, 3, 6, 11, 18, 28, 41, 59, 82, 112, 149, 196, 253, 323, 406, 507, 626, 768, 933, 1128, 1353, 1615, 1914, 2260, 2653, 3103, 3610, 4187, 4834, 5564, 6377, 7291, 8306, 9440, 10693, 12088, 13625, 15327, 17194, 19256, 21513, 23995, 26702, 29671, 32902, 36432

Offset: 1

Philippe Deléham, Jun 19 2010

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000123, A033485.

b:= proc(n) option remember;
      `if`(n<2, n, b(n-1)+b(iquo(n, 2)))
    end:
a:= n-> (b(2*n+1)-1)/2:
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 17 2022

from itertools import islice
from collections import deque
def A178855_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    while True:
        a += b
        aqueue.append(a)
        if f:
            yield (a-1)//2
            b = aqueue.popleft()
        f = not f
A178885_list = list(islice(A178855_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = (A033485(2n+1)-1)/2.

a(40) corrected by Georg Fischer, Aug 28 2020

A307148 Number of binary partitions of n in which exactly one of the powers of 2 is used an odd number of times.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 2, 5, 4, 7, 4, 10, 6, 12, 6, 17, 10, 21, 10, 28, 14, 32, 14, 42, 20, 48, 20, 60, 26, 66, 26, 83, 36, 93, 36, 114, 46, 124, 46, 152, 60, 166, 60, 198, 74, 212, 74, 254, 94, 274, 94, 322, 114, 342, 114, 402, 140, 428, 140, 494, 166, 520

Offset: 0

N. J. A. Sloane, Mar 28 2019

nonn

If someone extends this, the analogs L(m, n) = numbers of binary partitions of n in which exactly m of the powers of 2 are used an odd number of times for m>2 could also be added (A307149 is the case m=2).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
George E. Andrews and Jim Lawrence, Binary partitions and binary partition polytopes, preprint.
George E. Andrews and Jim Lawrence, Binary partitions and binary partition polytopes, Aequationes mathematicae 91.5 (2017): 859-869.

Cf. A000123, A307149.

Clear[L]; L[m_, n_] := L[m, n] = If[n == 0, If[m == 0, 1, 0], If[EvenQ[n] && n >= 2, L[m, n - 2] + L[m, n/2], If[m >= 1, L[m - 1, n - 1], 0]]]; Table[L[1, n], {n, 0, 100}] (* Vaclav Kotesovec, Mar 29 2019 *)

More terms from Vaclav Kotesovec, Mar 29 2019

A307149 Number of binary partitions of n in which exactly two of the powers of 2 are used an odd number of times.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 3, 1, 5, 3, 7, 4, 10, 7, 12, 8, 17, 13, 21, 16, 28, 23, 32, 27, 42, 37, 48, 44, 60, 56, 66, 64, 83, 81, 93, 94, 114, 115, 124, 131, 152, 159, 166, 182, 198, 214, 212, 241, 254, 283, 274, 320, 322, 368, 342, 412, 402, 472, 428, 528, 494

Offset: 0

N. J. A. Sloane, Mar 28 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
George E. Andrews and Jim Lawrence, Binary partitions and binary partition polytopes, preprint.
George E. Andrews and Jim Lawrence, Binary partitions and binary partition polytopes, Aequationes mathematicae 91.5 (2017): 859-869.

Cf. A000123, A307148.

Clear[L]; L[m_, n_] := L[m, n] = If[n == 0, If[m == 0, 1, 0], If[EvenQ[n] && n >= 2, L[m, n - 2] + L[m, n/2], If[m >= 1, L[m - 1, n - 1], 0]]]; Table[L[2, n], {n, 0, 100}] (* Vaclav Kotesovec, Mar 29 2019 *)

More terms from Vaclav Kotesovec, Mar 29 2019

A322003 Indices where A028897(A322000(n)) increases. Partial sums of A072170(n,10).

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 59, 72, 90, 108, 130, 152, 182, 212, 248, 284, 329, 374, 426, 478, 542, 606, 678, 750, 834, 918, 1011, 1104, 1214, 1324, 1446, 1568, 1708, 1848, 2002, 2156, 2333, 2510, 2702, 2894, 3108, 3322, 3552, 3782, 4040, 4298, 4575, 4852, 5156, 5460, 5784, 6108, 6464, 6820, 7196, 7572, 7977, 8382

Offset: 0

M. F. Hasler, Feb 19 2019

nonn

A322000 lists all nonnegative integers m ordered by increasing "decibinary" value N = A028897(m) = Sum d[i]*2^i where d[i] are the decimal digits of m. A072170(N,10) says in how many ways a given N can be written in that way. Accordingly, this is also the length of runs of identical values A028897(A322000(k)), and the partial sums, listed here as a(k), give the indices of A322000 where the decibinary value of the terms go up by one.

We have a(k) <= A000123(k-1) with equality for 1 <= k <= 10: the first differences of A000123 give back that sequence with terms duplicated, and this is the limiting column of A072170.

Cf. A322000, A028897, A072170.

A322003(n)=sum(k=0,n-1,A072170(k,10))
A322003_vec=vector(99,k,s=if(k>1,s)+A072170(k-1,10)) \\ more efficient for computing a large vector. Excludes the initial a(0) = 0 to have 1-based indices of the vector match the indices of the components a(n), n >= 1.

a(n) = Sum_{0 <= k < n} A072170(k,10).

A323775 a(n) = Sum_{k = 1...n} k^(2^(n - k)).

Original entry on oeis.org

1, 3, 8, 30, 359, 72385, 4338080222, 18448597098193762732, 340282370354622283774333836315916425069, 115792089237316207213755562747271079374483128445080168204415615259394085515423

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of ways to choose a constant integer partition of each part of a constant integer partition of 2^(n - 1).

			The a(1) = 1 through a(4) = 30 twice-partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (11)(2)       (22)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (1111)(4)
                             (4)(1111)
                             (11111111)
                             (1111)(22)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (11)(2)(2)(2)
                             (2)(11)(2)(2)
                             (2)(2)(11)(2)
                             (2)(2)(2)(11)
                             (11)(11)(2)(2)
                             (11)(2)(11)(2)
                             (11)(2)(2)(11)
                             (2)(11)(11)(2)
                             (2)(11)(2)(11)
                             (2)(2)(11)(11)
                             (11)(11)(11)(2)
                             (11)(11)(2)(11)
                             (11)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Cf. A000123, A001970, A002577, A006171, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323776.

Mathematica
```
Table[Sum[k^2^(n-k),{k,n}],{n,12}]
```

A073463 Triangle of number of partitions of 2n into powers of 2 where the largest part is 2^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 0, 1, 4, 4, 1, 0, 1, 5, 6, 2, 0, 0, 1, 6, 9, 4, 0, 0, 0, 1, 7, 12, 6, 0, 0, 0, 0, 1, 8, 16, 10, 1, 0, 0, 0, 0, 1, 9, 20, 14, 2, 0, 0, 0, 0, 0, 1, 10, 25, 20, 4, 0, 0, 0, 0, 0, 0, 1, 11, 30, 26, 6, 0, 0, 0, 0, 0, 0, 0, 1, 12, 36, 35, 10, 0, 0, 0, 0, 0, 0, 0, 0, 1, 13, 42, 44

Offset: 0

Henry Bottomley, Aug 02 2002

In the recurrence T(n,k)=T(n-1,k)+T([n/2],k-1): T(n-1,k) represents the partitions where the smallest part is 1 and T([n/2],k-1) those where it is not.

			Rows start:
  1;
  1, 1;
  1, 2, 1;
  1, 3, 2, 0;
  1, 4, 4, 1, 0;
  1, 5, 6, 2, 0, 0;
  ...

H. Bottomley, Illustration of initial terms

Columns include A000012, A000027, A002620, A008804. Subsequent columns start like A000123 (offset). Row sums are A000123.

T(n, k) = T(n-1, k)+T([n/2], k-1) starting with T(n, 0)=1 and T(0, k)=0 for k>0.

A101566 Binary partition sequence matrix.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 1, 4, 4, 2, 2, 1, 1, 6, 4, 4, 2, 2, 1, 1, 6, 6, 4, 4, 2, 2, 1, 1, 10, 6, 6, 4, 4, 2, 2, 1, 1, 10, 10, 6, 6, 4, 4, 2, 2, 1, 1, 14, 10, 10, 6, 6, 4, 4, 2, 2, 1, 1, 14, 14, 10, 10, 6, 6, 4, 4, 2, 2, 1, 1, 20, 14, 14, 10, 10, 6, 6, 4, 4, 2, 2, 1, 1, 20, 20, 14, 14, 10

Offset: 0

Paul Barry, Dec 07 2004

Row sums are A000123.

Cf. A018819.

Inverse matrix of T(n, k)=if(k<=n, (-1)^A010060(n-k), 0)

A117115 A sequence related to M-partitions.

Original entry on oeis.org

1, 2, 3, 6, 7, 11, 14, 21, 24, 34, 37, 51, 58, 75, 82, 110, 117, 148, 161, 202, 215, 265, 278, 342, 365, 436, 459, 557, 580, 685, 722, 855, 892, 1046, 1083, 1268, 1325, 1523, 1580, 1839, 1896, 2168, 2251, 2573, 2656, 3017, 3100, 3525, 3644, 4092, 4211, 4766, 4885

Offset: 0

N. J. A. Sloane, Apr 26 2006

nonn

O. J. Rodseth, Enumeration of M-partitions, Discrete Math., 306 (2006), 694-698. (See D(x).)

Cf. A000123, A100529, A117117.

# To get about 80 terms: B:=mul( (1-x^(2^n))^(-1),n=0..7); D2:=add( x^(2^(j-1)-1)*subs(x=x^(3*2^(j-1)),B)*mul(1/(1-x^(2^i)),i=0..j), j=1..8); series(D2,x,81);

A117117 A sequence related to M-partitions.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 13, 15, 21, 29, 37, 45, 62, 70, 89, 108, 132, 151, 191, 210, 256, 296, 350, 390, 476, 516, 610, 684, 795, 869, 1025, 1099, 1274, 1399, 1593, 1718, 1994, 2119, 2414, 2614, 2949, 3149, 3585, 3785, 4267, 4577, 5099, 5409, 6102, 6412, 7145, 7603, 8422

Offset: 0

N. J. A. Sloane, Apr 26 2006

nonn

O. J. Rodseth, Enumeration of M-partitions, Discrete Math., 306 (2006), 694-698. (See E(x).)

Cf. A000123, A100529, A117115.

# To get about 80 terms, first define B and D2 as in A117115. E2:=add( x^(2^(j+1)-4)*subs(x=x^(3*2^(j-1)),D2)*mul(1/(1-x^(2^i)),i=0..j), j=1..8); series(E2,x,81);

cp's OEIS Frontend

A162588 G.f.: A(x) = exp( 2*Sum_{n>=1} 2^n/A006519(n) * x^n/n ), where A006519(n) = highest power of 2 dividing n.

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A178855 Partial sums of A033485.

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A307148 Number of binary partitions of n in which exactly one of the powers of 2 is used an odd number of times.

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A307149 Number of binary partitions of n in which exactly two of the powers of 2 are used an odd number of times.

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Crossrefs

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Mathematica

Extensions

A322003 Indices where A028897(A322000(n)) increases. Partial sums of A072170(n,10).

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PARI

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A323775 a(n) = Sum_{k = 1...n} k^(2^(n - k)).

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Mathematica

A073463 Triangle of number of partitions of 2n into powers of 2 where the largest part is 2^k.

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A101566 Binary partition sequence matrix.

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A117115 A sequence related to M-partitions.

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A162588 G.f.: A(x) = exp( 2Sum_{n>=1} 2^n/A006519(n) x^n/n ), where A006519(n) = highest power of 2 dividing n.