A316245 -id:A316245 - cp's OEIS frontend

A133494 Diagonal of the array of iterated differences of A047848.

1, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883

Offset: 0

Paul Barry, Paul Curtz, Dec 23 2007

a(n) is the number of ways to choose a composition C, and then choose a composition of each part of C. - Geoffrey Critzer, Mar 19 2012
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the reptend length of 1/3^(n+1) in decimal. - Jianing Song, Nov 14 2018
Also the number of pairs of integer compositions, the first summing to n and the second with sum equal to the length of the first. If an integer composition is regarded as an arrow from sum to length, these are composable pairs, and the obvious composition operation founds a category of integer compositions. For example, we have (2,1,1,4) . (1,2,1) . (1,2) = (2,6), where dots represent the composition operation. The version without empty compositions is A000244. Composable triples are counted by 1 followed by A000302. The unordered version is A022811. - Gus Wiseman, Jul 14 2022

			From _Gus Wiseman_, Jul 15 2020: (Start)
The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
  ()  (1)  (2)      (3)
           (1,1)    (1,2)
           (1),(1)  (2,1)
                    (1,1,1)
                    (1),(2)
                    (2),(1)
                    (1),(1,1)
                    (1,1),(1)
                    (1),(1),(1)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (3).

The strict version is A336139.
Splittings of partitions are A323583.
Multiset partitions of partitions are A001970.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict compositions of each part of a composition are A307068.
Compositions of each part of a strict composition are A336127.
Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
Cf. A339006, A355382, A355384, A355387.

[n eq 0 select 1 else 3^(n-1): n in [0..30]]; // G. C. Greubel, Nov 20 2023

a:= n-> ceil(3^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2020

CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 3^(Range[0, 30])] (* G. C. Greubel, Nov 20 2023 *)

a(n)=max(1,3^(n-1)) \\ Charles R Greathouse IV, Jul 07 2011

Vec((1-2*x)/(1-3*x) + O(x^100)) \\  Altug Alkan, Oct 30 2015

[(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023

Binomial transform of A078008. - Paul Curtz, Aug 04 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: (1 - 2*x)/(1 - 3*x).
a(n) = A000244(n-1), n > 0. (End)
From Philippe Deléham, Nov 13 2008: (Start)
a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
Let A(x) be the g.f. Then B(x) = x*A(x) satisfies B(x/(1-x)) = x/(1 - 2*B(x)). - Vladimir Kruchinin, Dec 05 2011
G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - Joerg Arndt, Sep 30 2012
For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - J. Conrad, Oct 29 2015
G.f.: 1 + x/(1 + x)*(1 + 4*x/(1 + 4*x)*(1 + 7*x/(1 + 7*x)*(1 + 10*x/(1 + 10*x)*(1 + .... - Peter Bala, May 27 2017
Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - Gus Wiseman, Jul 19 2020
a(n) = ceiling(3^(n-1)). - Alois P. Heinz, Jul 26 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: (2 + exp(3*x))/3.
a(n) = 3*a(n-1) for n > 1. (End)

Definition clarified by R. J. Mathar, Nov 11 2008

A075900 Expansion of g.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n).

Original entry on oeis.org

1, 1, 3, 7, 19, 43, 115, 259, 659, 1523, 3731, 8531, 20883, 47379, 113043, 259219, 609683, 1385363, 3245459, 7344531, 17028499, 38579603, 88585619, 199845267, 457864595, 1028904339, 2339763603, 5256820115, 11896157587, 26626389395

Offset: 0

N. J. A. Sloane, Oct 15 2002

nonn

Number of compositions of partitions of n. a(3) = 7: 3, 21, 12, 111, 2|1, 11|1, 1|1|1. - Alois P. Heinz, Sep 16 2019
Also the number of ways to split an integer composition of n into consecutive subsequences with weakly decreasing (or increasing) sums. - Gus Wiseman, Jul 13 2020
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^(n-1). - Seiichi Manyama, Aug 22 2020

			From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 19 splittings:
  ()  (1)  (2)      (3)          (4)
           (1,1)    (1,2)        (1,3)
           (1),(1)  (2,1)        (2,2)
                    (1,1,1)      (3,1)
                    (2),(1)      (1,1,2)
                    (1,1),(1)    (1,2,1)
                    (1),(1),(1)  (2,1,1)
                                 (2),(2)
                                 (3),(1)
                                 (1,1,1,1)
                                 (1,1),(2)
                                 (1,2),(1)
                                 (2),(1,1)
                                 (2,1),(1)
                                 (1,1),(1,1)
                                 (1,1,1),(1)
                                 (2),(1),(1)
                                 (1,1),(1),(1)
                                 (1),(1),(1),(1)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..3180 (terms 0..1000 from Alois P. Heinz)
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Row sums of A327549.
The strict case is A304961.
Partitions of partitions are A001970.
Splittings with equal sums are A074854.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Splittings with distinct sums are A336127.
Starting with a reversed partition gives A316245.
Starting with a partition instead of composition gives A336136.
Cf. A006951, A063834, A300579, A317715, A319794, A323582, A327548, A336135.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(&*[1-2^(j-1)*x^j: j in [1..m+2]]) )); // G. C. Greubel, Jan 25 2024

oo := 101; t1 := mul(1/(1-x^n/2),n=1..oo): t2 := series(t1,x,oo-1): t3 := seriestolist(t2): A075900 := n->2^n*t3[n+1];
with(combinat); A075900 := proc(n) local i,t1,t2,t3; t1 := partition(n); t2 := 0; for i from 1 to nops(t1) do t3 := t1[i]; t2 := t2+2^(n-nops(t3)); od: t2; end;

b[n_]:= b[n]= Sum[d*2^(n - n/d), {d, Divisors[n]}];
a[0]= 1; a[n_]:= a[n]= 1/n*Sum[b[k]*a[n-k], {k,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Mar 20 2014, after Vladeta Jovovic, fixed by Vaclav Kotesovec, Mar 08 2018 *)

s(m,n):=if nVladimir Kruchinin, Sep 06 2014 */

{a(n)=polcoeff(prod(k=1,n,1/(1-2^(k-1)*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Jan 13 2013

{a(n)=polcoeff(exp(sum(k=1,n+1,x^k/(k*(1-2^k*x^k)+x*O(x^n)))),n)} \\ Paul D. Hanna, Jan 13 2013

m=80;
def A075900_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/product(1-2^(j-1)*x^j for j in range(1,m+1)) ).list()
A075900_list(m) # G. C. Greubel, Jan 25 2024

a(n) = Sum_{ partitions n = c_1 + ... + c_k } 2^(n-k). If p(n, m) = number of partitions of n into m parts, a(n) = Sum_{m=1..n} p(n, m)*2^(n-m).
G.f.: Sum_{n>=0} (a(n)/2^n)*x^n = Product_{n>0} 1/(1-x^n/2). - Vladeta Jovovic, Feb 11 2003
a(n) = 1/n*Sum_{k=1..n} A080267(k)*a(n-k). - Vladeta Jovovic, Feb 11 2003
G.f.: exp( Sum_{n>=1} x^n / (n*(1 - 2^n*x^n)) ). - Paul D. Hanna, Jan 13 2013
a(n) = s(1,n), a(0)=1, where s(m,n) = Sum_{k=m..n/2} 2^(k-1)*s(k, n-k)  + 2^(n-1), s(n,n) = 2^(n-1), s(m,n)=0, m>. - Vladimir Kruchinin, Sep 06 2014
a(n) ~ 2^(n-2) * (Pi^2 - 6*log(2)^2)^(1/4) * exp(sqrt((Pi^2 - 6*log(2)^2)*n/3)) / (3^(1/4) * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Mar 09 2018

More terms from Vladeta Jovovic, Feb 11 2003

A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, 1, 2, 6, 12, 32, 72, 176, 384, 960, 2112, 4992, 11264, 26112, 58368, 136192, 301056, 688128, 1548288, 3489792, 7766016, 17596416, 38993920, 87293952, 194248704, 432537600, 957349888, 2132803584, 4699717632, 10406068224, 23001563136, 50683969536, 111434268672, 245819768832

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Number of compositions of partitions of n into distinct parts. a(3) = 6: 3, 21, 12, 111, 2|1, 11|1. - Alois P. Heinz, Sep 16 2019
Also the number of ways to split a composition of n into contiguous subsequences with strictly decreasing sums. - Gus Wiseman, Jul 13 2020
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = (-1) * 2^(n-1). - Seiichi Manyama, Aug 22 2020

			From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 12 splittings:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (2),(1)    (1,1,2)
                  (1,1),(1)  (1,2,1)
                             (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1,1,1),(1)
(End)

Cf. A000009, A011782, A022629, A098407, A102866, A266964, A271619, A279785.
The non-strict version is A075900.
Starting with a reversed partition gives A323583.
Starting with a partition gives A336134.
Partitions of partitions are A001970.
Splittings with equal sums are A074854.
Splittings of compositions are A133494.
Splittings with distinct sums are A336127.
Cf. A006951, A063834, A316245, A317715, A319794, A323582, A336135, A336136.

nmax = 33; CoefficientList[Series[Product[(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+2^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020

G.f.: Product_{k>=1} (1 + A011782(k)*x^k).

a(n) ~ 2^n * exp(2*sqrt(-polylog(2, -1/2)*n)) * (-polylog(2, -1/2))^(1/4) / (sqrt(6*Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 19 2019

A317715 Number of ways to split an integer partition of n into consecutive subsequences with equal sums.

Original entry on oeis.org

1, 1, 3, 4, 9, 8, 21, 16, 39, 38, 64, 57, 146, 102, 186, 211, 352, 298, 593, 491, 906, 880, 1273, 1256, 2444, 1998, 3038, 3277, 4861, 4566, 7710, 6843, 10841, 10742, 14966, 15071, 24499, 21638, 31334, 32706, 47157, 44584, 67464, 63262, 91351, 94247, 125248

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(4) = 9 constant-sum split partitions:
  (4),
  (31),
  (22), (2)(2),
  (211), (2)(11),
  (1111), (11)(11), (1)(1)(1)(1).
The a(6) = 21 constant-sum split partitions:
  (6),
  (51),
  (42),
  (411),
  (33), (3)(3),
  (321), (3)(21),
  (3111), (3)(111),
  (222), (2)(2)(2),
  (2211), (2)(2)(11),
  (21111), (21)(111), (2)(11)(11),
  (111111), (111)(111), (11)(11)(11), (1)(1)(1)(1)(1)(1).

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..500
Gus Wiseman, The a(8) = 39 constant-sum split partitions.
Gus Wiseman, The a(10) = 64 constant-sum split partitions.
Gus Wiseman, The a(12) = 146 constant-sum split partitions.

Cf. A000041, A001970, A063834, A316223, A317545, A317546, A319001.

Cf. A316245, A317508, A318434, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[y],SameQ@@Total/@#&]],{y,IntegerPartitions[n]}],{n,10}]

a(16)-a(46) from Hiroaki Yamanouchi, Oct 02 2018

A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 25, 34, 54, 74, 109, 146, 211, 276, 381, 501, 675, 871, 1156, 1477, 1926, 2447, 3142, 3957, 5038, 6291, 7918, 9839, 12277, 15148, 18773, 23027, 28333, 34587, 42284, 51357, 62466, 75503, 91344, 109971, 132421, 158755, 190365, 227354, 271511

Offset: 0

Gus Wiseman, Jan 15 2019

nonn

			The a(5) = 14 split partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000005, A000219, A008284, A101509, A316245, A319066, A323295, A323300, A323307, A323429, A323434.

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, numtheory
      [tau](t+n), b(n, i-1, t)+b(n-i, min(n-i, i), t+1))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

Table[Sum[Length[Divisors[Length[ptn]]],{ptn,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1,
     DivisorSigma[0, t+n], b[n, i-1, t] + b[n-i, Min[n-i, i], t+1]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); Vec(1+sum(k=1, N, numdiv(k)*x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 21 2022

my(N=66, x='x+O('x^N)); Vec(1+sum(i=1, N, sum(j=1, N\i, x^(i*j)/prod(k=1, i*j, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022

a(n) = Sum_y A000005(k), where the sum is over all integer partitions of n and k is the number of parts.
From Seiichi Manyama, Jan 21 2022: (Start)
G.f.: 1 + Sum_{k>=1} A000005(k) * x^k/Product_{j=1..k} (1-x^j).
G.f.: 1 + Sum_{i>=1} Sum_{j>=1} x^(i*j)/Product_{k=1..i*j} (1-x^k). (End)
a(n) = Sum_{i=1..n} Sum_{j=1..n} A008284(n,i*j). - Ridouane Oudra, Apr 13 2023

A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 2, 8, 16, 48, 144, 352, 896, 2432, 7168, 16896, 46080, 114688, 303104, 843776, 2080768, 5308416, 13762560, 34865152, 87818240, 241172480, 583008256, 1503657984, 3762290688, 9604956160, 23689428992, 60532195328, 156397207552, 385137770496, 967978254336

Offset: 0

Gus Wiseman, Jul 09 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(4) = 16 splits:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (1),(2)    (1,1,2)
                  (2),(1)    (1,2,1)
                  (1),(1,1)  (1),(3)
                  (1,1),(1)  (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1),(1,2)
                             (1),(2,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1),(1,1,1)
                             (1,1,1),(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A074854.
Starting with a strict composition gives A336128.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(n) = Sum_{k=0..n} 2^(n-k) k! A008289(n,k).

A318684 Number of ways to split a strict integer partition of n into consecutive subsequences with strictly decreasing sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 11, 14, 20, 28, 35, 48, 61, 79, 105, 129, 162, 208, 257, 318, 404, 489, 600, 732, 896, 1075, 1315, 1576, 1895, 2272, 2715, 3217, 3851, 4537, 5377, 6353, 7484, 8765, 10314, 12044, 14079, 16420, 19114, 22184, 25818, 29840, 34528, 39903, 46030

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(9) = 20 split partitions:
    (9)
   (81)   (8)(1)
   (72)   (7)(2)
   (63)   (6)(3)
   (54)   (5)(4)
  (432)  (43)(2)  (4)(3)(2)
  (621)  (62)(1)  (6)(2)(1)  (6)(21)
  (531)  (53)(1)  (5)(3)(1)  (5)(31)

Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[y],OrderedQ[Total/@#,Greater]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A319794 Number of ways to split a strict integer partition of n into consecutive subsequences with weakly decreasing sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 20, 31, 37, 52, 64, 85, 111, 141, 175, 225, 279, 346, 437, 532, 654, 802, 979, 1182, 1438, 1740, 2083, 2502, 2996, 3565, 4245, 5043, 5950, 7068, 8303, 9772, 11449, 13452, 15681, 18355, 21338, 24855, 28846, 33509, 38687, 44819, 51644

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(6) = 9 split partitions:
    (6)
   (51)  (5)(1)
   (42)  (4)(2)
  (321)  (32)(1)  (3)(21)  (3)(2)(1).

Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A318684.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[y],OrderedQ[Total/@#,GreaterEqual]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

A074854 a(n) = Sum_{d|n} (2^(n-d)).

Original entry on oeis.org

1, 3, 5, 13, 17, 57, 65, 209, 321, 801, 1025, 3905, 4097, 12417, 21505, 53505, 65537, 233985, 262145, 885761, 1327105, 3147777, 4194305, 16060417, 17825793, 50339841, 84148225, 220217345, 268435457, 990937089, 1073741825, 3506503681

Offset: 1

Miklos Kristof, Sep 11 2002

A034729 = Sum_{d|n} (2^(d-1)).
A055895 = 2*A034729.
If p is a prime, then a(p) = A034729(p) = 2^(p-1)+1.
From Gus Wiseman, Jul 14 2020: (Start)
Number of ways to tile a rectangle of size n using horizontal strips. Also the number of ways to choose a composition of each part of a constant partition of n. The a(0) = 1 through a(5) = 17 splittings are:
  ()  (1)  (2)      (3)          (4)              (5)
           (1,1)    (1,2)        (1,3)            (1,4)
           (1),(1)  (2,1)        (2,2)            (2,3)
                    (1,1,1)      (3,1)            (3,2)
                    (1),(1),(1)  (1,1,2)          (4,1)
                                 (1,2,1)          (1,1,3)
                                 (2,1,1)          (1,2,2)
                                 (2),(2)          (1,3,1)
                                 (1,1,1,1)        (2,1,2)
                                 (1,1),(2)        (2,2,1)
                                 (2),(1,1)        (3,1,1)
                                 (1,1),(1,1)      (1,1,1,2)
                                 (1),(1),(1),(1)  (1,1,2,1)
                                                  (1,2,1,1)
                                                  (2,1,1,1)
                                                  (1,1,1,1,1)
                                                  (1),(1),(1),(1),(1)
(End)

			Divisors of 6 = 1,2,3,6 and 6-1 = 5, 6-2 = 4, 6-3 = 3, 6-6 = 0. a(6) = 2^5 + 2^4 + 2^3 + 2^0 = 32 + 16 + 8 + 1 = 57.
G.f. = x + 3*x^2 + 5*x^3 + 13*x^4 + 17*x^5 + 57*x^6 + 65*x^7 + ...
a(14) = 1 + 2^7 + 2^12 + 2^13 = 12417. - _Gus Wiseman_, Jun 20 2018

Gus Wiseman, Table of n, a(n) for n = 1..32

Cf. A055895, A034729.
Cf. A080267.
Cf. A051731.
The version looking at lengths instead of sums is A101509.
The strictly increasing (or strictly decreasing) version is A304961.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Requiring distinct instead of equal sums gives A336127.
Starting with a strict composition gives A336130.
Partitions of partitions are A001970.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Cf. A006951, A063834, A075900, A279787, A305551, A316245, A323433, A336128.

a[ n_] := If[ n < 1, 0, Sum[ 2^(n - d), {d, Divisors[n]}]] (* Michael Somos, Mar 28 2013 *)

a(n)=if(n<1,0,2^n*polcoeff(sum(k=1,n,2/(2-x^k),x*O(x^n)),n))

a(n) = sumdiv(n,d, 2^(n-d) ); /* Joerg Arndt, Mar 28 2013 */

G.f.: 2^n times coefficient of x^n in Sum_{k>=1} x^k/(2-x^k). - Benoit Cloitre, Apr 21 2003; corrected by Joerg Arndt, Mar 28 2013
G.f.: Sum_{k>0} 2^(k-1)*x^k/(1-2^(k-1)*x^k). - Vladeta Jovovic, Jun 24 2003
G.f.: Sum_{n>=1} a*z^n/(1-a*z^n) (generalized Lambert series) where z=2*x and a=1/2. - Joerg Arndt, Jan 30 2011
Triangle A051731 mod 2 converted to decimal. - Philippe Deléham, Oct 04 2003
G.f.: Sum_{k>0} 1 / (2 / (2*x)^k - 1). - Michael Somos, Mar 28 2013

a(14) corrected from 9407 to 12417 by Gus Wiseman, Jun 20 2018

A318683 Number of ways to split a strict integer partition of n into consecutive subsequences with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 5, 7, 8, 12, 12, 18, 18, 26, 27, 37, 38, 53, 54, 73, 76, 100, 104, 136, 142, 183, 192, 244, 256, 327, 340, 424, 448, 558, 585, 722, 760, 937, 983, 1195, 1260, 1544, 1610, 1943, 2053, 2480, 2590, 3107, 3264, 3927, 4106, 4874, 5120, 6134, 6378

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(12) = 18 constant-sum split partitions:
  (12)
  (7,5)
  (8,4)
  (9,3)
  (10,2)
  (11,1)
  (5,4,3)
  (6,4,2)
  (6,5,1)
  (7,3,2)
  (7,4,1)
  (8,3,1)
  (9,2,1)
  (6)(4,2)
  (6)(5,1)
  (5,4,2,1)
  (6,3,2,1)
  (6)(3,2,1)

Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[y],SameQ@@Total/@#&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

cp's OEIS Frontend

A133494 Diagonal of the array of iterated differences of A047848.

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A075900 Expansion of g.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n).

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A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

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A317715 Number of ways to split an integer partition of n into consecutive subsequences with equal sums.

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A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

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A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

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