A000123 -id:A000123 - cp's OEIS frontend

A172288 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^2^n into powers of 2 less than or equal to 2^k.

1, 2, 1, 2, 3, 1, 2, 4, 9, 1, 2, 4, 25, 129, 1, 2, 4, 35, 4225, 32769, 1, 2, 4, 36, 47905, 268468225, 2147483649, 1, 2, 4, 36, 222241, 733276217345, 1152921506754330625, 9223372036854775809, 1

Offset: 0

Alois P. Heinz, Jan 26 2011

A(18,18) = 2797884726...4715787265 has 1420371 decimal digits and was computed by the algorithm given below.

			A(2,1) = 9, because there are 9 partitions of 2^2^2=16 into powers of 2 less than or equal to 2^1=2: [2,2,2,2,2,2,2,2], [2,2,2,2,2,2,2,1,1], [2,2,2,2,2,2,1,1,1,1], [2,2,2,2,2,1,1,1,1,1,1], [2,2,2,2,1,1,1,1,1,1,1,1], [2,2,2,1,1,1,1,1,1,1,1,1,1], [2,2,1,1,1,1,1,1,1,1,1,1,1,1], [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
  1,     2,         2,            2,               2,  ...
  1,     3,         4,            4,               4,  ...
  1,     9,        25,           35,              36,  ...
  1,   129,      4225,        47905,          222241,  ...
  1, 32769, 268468225, 733276217345, 751333186150401,  ...

Alois P. Heinz, Table of n, a(n) for n = 0..77

Cf. A002577, A000123, A181322, A145515.

Main diagonal gives: A182135.

b:= proc(n,j) option remember; local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n b(2^(2^n-k), k):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[n_, j_] := b[n, j] = Module[{nn, r}, Which[n < 0, 0, j == 0, 1, j == 1, n+1, n < j , b[n, j] = b[n-1, j] + b[2*n, j-1] , True, nn = 1 + Floor[n]; r := n - nn; (nn-j)*Binomial[nn, j] * Sum [Binomial[j, h] /(nn - j + h) * b[j - h + r, j] *(-1)^h, {h, 0, j-1}] ] ]; a[n_, k_] := b[2^(2^n-k), k]; Table[Table[a[n, d-n] // FullSimplify, {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

A(n,k) = [x^2^(2^n-1)] 1/(1-x) * 1/Product_{j=0..k-1} (1-x^(2^j)).

A183036 G.f.: exp( Sum_{n>=1} A001511(n)2^A001511(n)x^n/n ) where A001511(n) equals the 2-adic valuation of 2n.

Original entry on oeis.org

1, 2, 6, 10, 24, 38, 74, 110, 200, 290, 486, 682, 1096, 1510, 2314, 3118, 4650, 6182, 8946, 11710, 16616, 21522, 29886, 38250, 52328, 66406, 89394, 112382, 149496, 186610, 245086, 303562, 394814, 486066, 625686, 765306, 977112, 1188918, 1504954

Offset: 0

Paul D. Hanna, Dec 19 2010

nonn

Compare to B(x), the g.f. of the binary partitions (A000123):
B(x) = exp( Sum_{n>=1} 2^A001511(n)*x^n/n ) = (1-x)^(-1)*Product_{n>=0} 1/(1 - x^(2^n)).
2^A001511(n) exactly divides 2n.

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 24*x^4 + 38*x^5 + 74*x^6 +...
log(A(x)) = 2*x + 8*x^2/2 + 2*x^3/3 + 24*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 64*x^8/8 + 2*x^9/9 + 8*x^10/10 +...+ A183037(n)*x^n/n +...

Cf. A183037, A183038, A001511, A000123.

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

G.f. satisfies: A(x) = (1-x^2)/(1-x)^2 * A(x^2)^2/A(x^4).

A187821 Number of non-squashing partitions of n into odd parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 9, 7, 9, 11, 12, 11, 12, 15, 17, 15, 17, 21, 22, 21, 22, 27, 29, 27, 29, 36, 36, 36, 36, 45, 47, 45, 47, 57, 58, 57, 58, 69, 73, 69, 73, 86, 88, 86, 88, 103, 109, 103, 109, 125, 130, 125, 130, 147, 157, 147, 157, 176, 184, 176, 184, 205, 220, 205, 220, 241, 256, 241, 256

Offset: 0

Joerg Arndt, Dec 27 2012

nonn

A non-squashing partition of n is a partition p(1) + p(2) + ... + p(m) = n such that p(k) >= sum(j=k+1..m, p(j) ).

			The a(33) = a(35) = 27 non-squashing partitions of 33 and 35 into odd parts are
[ 1]   [ 17 9 5 1 1 ]       [ 1]   [ 19 9 5 1 1 ]
[ 2]   [ 17 9 7 ]           [ 2]   [ 19 9 7 ]
[ 3]   [ 17 11 3 1 1 ]      [ 3]   [ 19 11 3 1 1 ]
[ 4]   [ 17 11 5 ]          [ 4]   [ 19 11 5 ]
[ 5]   [ 17 13 3 ]          [ 5]   [ 19 13 3 ]
[ 6]   [ 17 15 1 ]          [ 6]   [ 19 15 1 ]
[ 7]   [ 19 7 5 1 1 ]       [ 7]   [ 21 7 5 1 1 ]
[ 8]   [ 19 7 7 ]           [ 8]   [ 21 7 7 ]
[ 9]   [ 19 9 3 1 1 ]       [ 9]   [ 21 9 3 1 1 ]
[10]   [ 19 9 5 ]           [10]   [ 21 9 5 ]
[11]   [ 19 11 3 ]          [11]   [ 21 11 3 ]
[12]   [ 19 13 1 ]          [12]   [ 21 13 1 ]
[13]   [ 21 7 3 1 1 ]       [13]   [ 23 7 3 1 1 ]
[14]   [ 21 7 5 ]           [14]   [ 23 7 5 ]
[15]   [ 21 9 3 ]           [15]   [ 23 9 3 ]
[16]   [ 21 11 1 ]          [16]   [ 23 11 1 ]
[17]   [ 23 5 3 1 1 ]       [17]   [ 25 5 3 1 1 ]
[18]   [ 23 5 5 ]           [18]   [ 25 5 5 ]
[19]   [ 23 7 3 ]           [19]   [ 25 7 3 ]
[20]   [ 23 9 1 ]           [20]   [ 25 9 1 ]
[21]   [ 25 5 3 ]           [21]   [ 27 5 3 ]
[22]   [ 25 7 1 ]           [22]   [ 27 7 1 ]
[23]   [ 27 3 3 ]           [23]   [ 29 3 3 ]
[24]   [ 27 5 1 ]           [24]   [ 29 5 1 ]
[25]   [ 29 3 1 ]           [25]   [ 31 3 1 ]
[26]   [ 31 1 1 ]           [26]   [ 33 1 1 ]
[27]   [ 33 ]               [27]   [ 35 ]

Joerg Arndt, Table of n, a(n) for n = 0..1000

Cf. A018819 and A000123 (non-squashing partitions, also binary partitions).

Cf. A088567 (non-squashing partitions into distinct parts)

A217553 G.f.: exp( Sum_{n>=1} 4^A001511(n) * x^n/n ), where 2^A001511(n) is the highest power of 2 that divides 2*n.

Original entry on oeis.org

1, 4, 16, 44, 128, 308, 752, 1628, 3584, 7268, 14864, 28556, 55296, 102036, 189168, 337084, 603136, 1044676, 1814288, 3064556, 5188352, 8578548, 14205936, 23041308, 37420800, 59680548, 95265552, 149620812, 235161216, 364301652, 564627952, 863725948, 1321756672

Offset: 0

Paul D. Hanna, Oct 30 2012

nonn

Compare g.f. to the g.f. of binary partitions (A000123):

exp( Sum_{n>=1} 2^A001511(n) * x^n/n ).

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 44*x^3 + 128*x^4 + 308*x^5 + 752*x^6 +...
where
log(A(x)) = 4^1*x + 4^2*x^2/2 + 4^1*x^3/3 + 4^4*x^4/4 + 4^1*x^5/5 + 4^2*x^6/6 + 4^1*x^7/7 + 4^4*x^8/8 + 4^1*x^9/9 + 4^2*x^10/10 + 4^1*x^11/11 + 4^4*x^12/12 +...+ 4^A001511(n)*x^n/n +...

Cf. A162581, A180591, A001511.

{a(n)=polcoeff(exp(sum(m=1,n,4^valuation(2*m,2)*x^m/m)+x*O(x^n)),n)}
for(n=0,31,print1(a(n),", "))

Self-convolution of A162581.

A281488 a(n) = -Sum_{d divides (n-2), 1 <= d < n} a(d).

Original entry on oeis.org

1, -1, -1, 0, 0, 0, -1, 1, 0, -1, 0, 1, -1, 0, 0, 1, 0, -2, -1, 3, 0, -2, 1, 2, -2, -3, 1, 4, -1, -3, 0, 5, -1, -7, 1, 7, -1, -5, 0, 6, 1, -9, -2, 11, 1, -9, -1, 8, 0, -12, 0, 15, 0, -11, -1, 13, 0, -17, 1, 18, -2, -17, 1, 17, 0, -24, 0, 28, -1, -21, 0, 22

Offset: 1

Andrey Zabolotskiy, Jan 22 2017

a(1) = 1, any other choice simply adds a factor to all terms.

The even bisection of the sequence seems to behave similarly to A281487 with similar asymptotics for |a(n)|. However, the odd bisection shows oscillations with increasing intervals between crossing the zero and increasing amplitude.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..20000
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, Involve, 15 (2022), 251-270; arXiv preprint, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007439 (same formula with overall + instead of -), A281487 (same formula with (n-1) instead of (n-2)), A000123.

a = [1]
for n in range(2, 100):
   a.append(-sum(a[d-1] for d in range(1, n) if (n-2)%d == 0))
print(a)

a(1) = 1, a(n) = -Sum_{d|(n-2), 1 <= d < n} a(d) for n>1.

A289842 Sum of products of terms in all partitions of 2*n into powers of 2.

Original entry on oeis.org

1, 3, 11, 27, 83, 195, 515, 1155, 2899, 6387, 15219, 32883, 76275, 163059, 368883, 780531, 1738259, 3653715, 8022355, 16759635, 36428371, 75765843, 163217491, 338120787, 723384915, 1493913171, 3176799827, 6542573139, 13844246099, 28447592019, 59934789203

Offset: 0

Seiichi Manyama, Oct 27 2017

nonn

			n | partitions of 2*n into powers of 2                 | a(n)
--------------------------------------------------------------------------
1 | 2  , 1+1                                           | 2+1         =  3.
2 | 4  , 2+2  , 2+1+1, 1+1+1+1                         | 4+4+2+1     = 11.
3 | 4+2, 4+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1 | 8+4+8+4+2+1 = 27.

Seiichi Manyama, Table of n, a(n) for n = 0..3309

Cf. A000123, A018819.

b:= proc(n, i, p) option remember; `if`(n=0, p,
     `if`(i<1, 0, add(b(n-j*i, i/2, p*i^j), j=0..n/i)))
    end:
a:= n-> (t-> b(t, 2^ilog2(t), 1))(2*n):
seq(a(n), n=0..33);  # Alois P. Heinz, Oct 27 2017

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p, If[i < 1, 0, Sum[b[n - j i, i/2, p i^j], {j, 0, n/i}]]];
a[n_] := b[2n, 2^Floor@Log[2, 2n], 1];
a /@ Range[0, 33] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

a(n) = [x^(2*n)] Product_{k>=0} 1/(1 - 2^k*x^(2^k)). - Ilya Gutkovskiy, Sep 10 2018

a(n) ~ c * n * 2^n, where c = 2.1343755406794500897789546611306737041750472866941557748356... - Vaclav Kotesovec, Jun 18 2019

A303666 Expansion of 1/((1 - x)*(1 - Sum_{k>=0} x^(2^k))).

Original entry on oeis.org

1, 2, 4, 7, 13, 23, 41, 72, 128, 226, 400, 706, 1248, 2204, 3894, 6877, 12149, 21459, 37907, 66957, 118275, 208919, 369037, 651863, 1151453, 2033921, 3592719, 6346167, 11209863, 19801075, 34976589, 61782572, 109132628, 192771658, 340511506, 601478868, 1062451154, 1876711698, 3315020026

Offset: 0

Ilya Gutkovskiy, Apr 28 2018

nonn

Partial sums of A023359.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Index entries for sequences related to compositions

Cf. A000079, A000123, A023359, A209229.

a:= proc(n) option remember; 1+
     `if`(n>0, add(a(n-2^i), i=0..ilog2(n)), 0)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 28 2018

nmax = 38; CoefficientList[Series[1/((1 - x) (1 - Sum[x^2^k, {k, 0, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[k == 2^IntegerExponent[k, 2]] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 38}]]

A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

Original entry on oeis.org

1, 3, 7, 16, 40, 119, 450, 2253, 15207, 139190, 1731703, 29335875, 677864041, 21400069232, 924419728471, 54716596051100, 4443400439075834, 495676372493566749, 76041424515817042402, 16060385520094706930608, 4674665948889147697184915

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of multiset partitions of integer partitions of 2^(n - 1) whose parts are constant and have equal sums.

			The a(1) = 1 through a(4) = 16 partitions of partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (4)(1111)
                             (11111111)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (2)(2)(2)(11)
                             (2)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Seiichi Manyama, Table of n, a(n) for n = 1..120

Cf. A000123, A001970, A002577, A006171, A007716, A034729, A047968, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323775.

Table[Sum[Binomial[k+2^(n-k)-1,k-1],{k,n}],{n,20}]

a(n) = sum(k=1, n, binomial(k+2^(n-k)-1, k-1)); \\ Michel Marcus, Jan 28 2019

A342247 Number of partitions of n into seven powers of 2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 3, 4, 4, 4, 4, 5, 4, 6, 5, 6, 6, 7, 5, 8, 6, 7, 6, 8, 5, 7, 5, 7, 6, 9, 6, 9, 7, 9, 7, 11, 8, 10, 7, 10, 7, 10, 6, 11, 8, 10, 7, 12, 7, 10, 7, 11, 7, 10, 5, 9, 5, 8, 5, 10, 7, 10, 6, 11, 9, 12, 8, 14, 9, 11, 7, 13, 8, 12, 8, 14, 10, 13, 8, 15, 9, 13

Offset: 7

Ilya Gutkovskiy, Mar 07 2021

nonn

Column k=7 of A089052.

Cf. A000079, A000123, A018819, A075897, A089048, A089049, A089050, A089051, A209229, A319922, A342248, A342249, A342250.

A342248 Number of partitions of n into eight powers of 2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 4, 5, 5, 6, 6, 7, 6, 8, 7, 8, 8, 10, 7, 10, 8, 9, 7, 10, 7, 10, 9, 11, 9, 12, 9, 13, 11, 14, 10, 14, 10, 13, 10, 14, 11, 15, 10, 15, 12, 15, 10, 17, 11, 14, 10, 15, 9, 13, 8, 14, 10, 14, 10, 16, 11, 16, 12, 18, 14, 18, 11, 18, 13, 17, 12, 20

Offset: 8

Ilya Gutkovskiy, Mar 07 2021

nonn

Column k=8 of A089052.

Cf. A000079, A000123, A018819, A075897, A089048, A089049, A089050, A089051, A209229, A319922, A342247, A342249, A342251.

cp's OEIS Frontend

A172288 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^2^n into powers of 2 less than or equal to 2^k.

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A183036 G.f.: exp( Sum_{n>=1} A001511(n)*2^A001511(n)*x^n/n ) where A001511(n) equals the 2-adic valuation of 2n.

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A187821 Number of non-squashing partitions of n into odd parts.

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A217553 G.f.: exp( Sum_{n>=1} 4^A001511(n) * x^n/n ), where 2^A001511(n) is the highest power of 2 that divides 2*n.

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A281488 a(n) = -Sum_{d divides (n-2), 1 <= d < n} a(d).

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A289842 Sum of products of terms in all partitions of 2*n into powers of 2.

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Maple

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A303666 Expansion of 1/((1 - x)*(1 - Sum_{k>=0} x^(2^k))).

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A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

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A183036 G.f.: exp( Sum_{n>=1} A001511(n)2^A001511(n)x^n/n ) where A001511(n) equals the 2-adic valuation of 2n.