A018819 -id:A018819 - cp's OEIS frontend

A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

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

Offset: 1

Ilya Gutkovskiy, Nov 21 2020

nonn

From Gus Wiseman, Mar 05 2021: (Start)
This sequence counts all of the following essentially equivalent things:
1. Chains of distinct inferior divisors from n to 1, where a divisor d|n is inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
2. Chains of divisors from n to 1 whose first-quotients (in analogy with first-differences) are term-wise greater than or equal to their decapitation (maximum element removed). For example, the divisor chain q = 60/4/2/1 has first-quotients (15,2,2), which are >= (4,2,1), so q is counted under a(60).
3. Chains of divisors from n to 1 such that x >= y^2 for all adjacent x, y.
4. Factorizations of n where each factor is greater than or equal to the product of all previous factors.
(End)

			From _Gus Wiseman_, Mar 05 2021: (Start)
The a(n) chains for n = 1, 2, 4, 12, 16, 24, 36, 60:
  1  2/1  4/1    12/1    16/1      24/1      36/1      60/1
          4/2/1  12/2/1  16/2/1    24/2/1    36/2/1    60/2/1
                 12/3/1  16/4/1    24/3/1    36/3/1    60/3/1
                         16/4/2/1  24/4/1    36/4/1    60/4/1
                                   24/4/2/1  36/6/1    60/5/1
                                             36/4/2/1  60/6/1
                                             36/6/2/1  60/4/2/1
                                                       60/6/2/1
The a(n) factorizations for n = 2, 4, 12, 16, 24, 36, 60:
    2  4    12   16     24     36     60
       2*2  2*6  2*8    3*8    4*9    2*30
            3*4  4*4    4*6    6*6    3*20
                 2*2*4  2*12   2*18   4*15
                        2*2*6  3*12   5*12
                               2*2*9  6*10
                               2*3*6  2*2*15
                                      2*3*10
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A002033, A008578 (positions of 1's), A068108.
The restriction to powers of 2 is A018819.
Not requiring inferiority gives A074206 (ordered factorizations).
The strictly inferior version is A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A000929, or A342098 forbidding equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts strict factorizations of divisors.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A001248, A006530, A020639, A040039, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

a:= proc(n) option remember; `if`(n=1, 1, add(
      `if`(d<=n/d, a(d), 0), d=numtheory[divisors](n)))
    end:
seq(a(n), n=1..128);  # Alois P. Heinz, Jun 24 2021

a[1] = 1; a[n_] := a[n] = DivisorSum[n, a[#] &, # <= Sqrt[n] &]; Table[a[n], {n, 95}]
(* second program *)
asc[n_]:=Prepend[#,n]&/@Prepend[Join@@Table[asc[d],{d,Select[Divisors[n],#Gus Wiseman, Mar 05 2021 *)

G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^k).

a(2^n) = A018819(n). - Gus Wiseman, Mar 08 2021

A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 4, 8, 2, 10, 2, 6, 6, 4, 2, 12, 2, 6, 4, 6, 2, 10, 4, 8, 4, 4, 2, 14, 2, 4, 6, 6, 4, 10, 2, 6, 4, 8, 2, 16, 2, 4, 6, 6, 4, 10, 2, 12, 4, 4, 2, 14

Offset: 1

Gus Wiseman, Mar 05 2021

nonn

An alternative wording: Number of chains of divisors starting with n and having all adjacent parts x > y^2.

			The chains for n = 1, 2, 6, 12, 24, 42, 48:
   1    2      6        12        24        42          48
        2/1    6/1      12/1      24/1      42/1        48/1
               6/2      12/2      24/2      42/2        48/2
               6/2/1    12/3      24/3      42/3        48/3
                        12/2/1    24/4      42/6        48/4
                        12/3/1    24/2/1    42/2/1      48/6
                                  24/3/1    42/3/1      48/2/1
                                  24/4/1    42/6/1      48/3/1
                                            42/6/2      48/4/1
                                            42/6/2/1    48/6/1
                                                        48/6/2
                                                        48/6/2/1

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A018819.
Not requiring strict inferiority gives A067824.
The weakly inferior version is twice A337135.
The case ending with 1 is counted by A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A342098, or A000929 allowing equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A334997 counts chains of divisors of n by length.
Cf. A002033, A006530, A018819, A020639, A337105, A341674, A342086, A342094, A342095, A342097.

cem[n_]:=Prepend[Prepend[#,n]&/@Join@@cem/@Most[Divisors[n]],{n}];
Table[Length[Select[cem[n],And@@Thread[Divide@@@Partition[#,2,1]>Rest[#]]&]],{n,30}]

For n > 1, a(n) = 2*A342083(n).

A350844 Number of strict integer partitions of n with no difference -2.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 4, 4, 7, 7, 8, 11, 12, 15, 18, 21, 23, 31, 32, 40, 45, 54, 59, 73, 78, 94, 106, 122, 136, 161, 177, 203, 231, 259, 293, 334, 372, 417, 476, 525, 592, 663, 742, 821, 931, 1020, 1147, 1271, 1416, 1558, 1752, 1916, 2137, 2357, 2613, 2867

Offset: 0

Gus Wiseman, Jan 21 2022

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4   5    6     7    8     9     A      B     C
          21       32   51    43   62    54    73     65    84
                   41   321   52   71    63    82     74    93
                              61   521   72    91     83    A2
                                         81    541    92    B1
                                         432   721    A1    543
                                         621   4321   632   651
                                                      821   732
                                                            741
                                                            921
                                                            6321

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for no difference 0 is A000009.
The version for no difference > -2 is A001227, non-strict A034296.
The version for no difference -1 is A003114 (A325160).
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The opposite version is A072670.
The multiplicative version is A350840, non-strict A350837 (A350838).
The non-strict version is A350842.
A000041 counts integer partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length (A026424).
A116931 counts partitions with no difference -1 (A319630).
A323092 counts double-free integer partitions (A320340) strict A120641.
A325534 counts separable partitions (A335433).
A325535 counts inseparable partitions (A335448).
Cf. A000929, A003000, A018819, A040039, A045690, A045691, A154402, A303362, A323094, A342095, A342097.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],0|-2]&]],{n,0,30}]

A002449 Number of different types of binary trees of height n.

Original entry on oeis.org

1, 1, 2, 6, 26, 166, 1626, 25510, 664666, 29559718, 2290267226, 314039061414, 77160820913242, 34317392762489766, 27859502236825957466, 41575811106337540656038, 114746581654195790543205466, 588765612737696531880325270438, 5642056933026209681424588087899226

Offset: 0

N. J. A. Sloane

Two trees have the same type if they have the same number of nodes at each level. - Chams Lahlou, Jan 26 2019

Equals the number of partitions of 2^n-1 into powers of 2 (cf. A018819). a(n) = A018819(2^n-1) = binary partitions of 2^n-1. - Paul D. Hanna, Sep 22 2004

			G.f. = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 166*x^5 + 1626*x^6 + 25510*x^7 + ...

George E. Andrews, Peter Paule, Axel Riese and Volker Strehl, "MacMahon's Partition Analysis V: Bijections, recursions and magic squares," in Algebraic Combinatorics and Applications, edited by Anton Betten, Axel Kohnert, Reinhard Laue and Alfred Wassermann [Proceedings of ALCOMA, September 1999] (Springer, 2001), 1-39.
A. Cayley, "On a problem in the partition of numbers," Philosophical Magazine (4) 13 (1857), 245-248; reprinted in his Collected Math. Papers, Vol. 3, pp. 247-249. - Don Knuth, Aug 17 2001
R. F. Churchhouse, Congruence properties of the binary partition function. Proc. Cambridge Philos. Soc. 66 1969 371-376.
R. F. Churchhouse, Binary partitions, pp. 397-400 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
D. E. Knuth, Selected Papers on Analysis of Algorithms, p. 75 (gives asymptotic formula and lower bound).
H. Minc, The free commutative entropic logarithmetic. Proc. Roy. Soc. Edinburgh Sect. A 65 1959 177-192 (1959).
T. K. Moon (tmoon(AT)artemis.ece.usu.edu), Enumerations of binary trees, types of trees and the number of reversible variable length codes, submitted to Discrete Applied Mathematics, 2000.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
A. Cayley, On a problem in the partition of numbers, Philosophical Magazine (4) 13 (1857), 245-248; reprinted in his Collected Math. Papers, Vol. 3, pp. 247-249.
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See pp. 31, 33, 34.
Chams Lahlou, A formula for some integer sequences that can be described by generating trees

Cf. A001699, A056207, A098539, A018819.

d := proc(n) option remember; if n<1 then 1 else sum(d(n-1),k=1..2*k) fi end; A002449 := n -> eval(d(n-1),k=1); # Michael Kleber, Dec 05 2000

lim = 16; p[0] = p[1] = 1; Do[If[OddQ[n], p[n] = p[n - 1], p[n] = p[n - 1] + p[n/2]], {n, 1, 2^lim - 1}]; a[n_] := p[2^n - 1]; Table[a[n], {n, 0, lim}] (* Jean-François Alcover, Sep 20 2011, after Paul D. Hanna *)

a(n)=local(A,B,C,m);A=matrix(1,1);A[1,1]=1; for(m=2,n+1,B=A^2;C=matrix(m,m);for(j=1,m, for(k=1,j, if(j<3 || k==j || k>m-1,C[j,k]=1,if(k==1,C[j,k]=B[j-1,1],C[j,k]=B[j-1,k-1])); ));A=C);A[n+1,1] \\ Paul D. Hanna

a(n)=polcoeff(1/prod(k=0,n,1-x^(2^k)+O(x^(2^n))),2^n-1)

{a(n, k=2) = if(n<2, n>=0, sum(i=1, k, a(n-1, 2*i)))}; /* Michael Somos, Nov 24 2016 */

a(n) = A098539(n, 1). - Paul D. Hanna, Sep 13 2004
G.f. A(x) = F(x,1) where F(x,n) satisfies: F(x,n) = F(x,n-1) + xF(x,2n) for n>0 with F(x,0)=1. - Paul D. Hanna, Apr 16 2007
From Benedict W. J. Irwin, Nov 16 2016: (Start)
Conjecture: a(n+2) = Sum_{i_1=1..2}Sum_{i_2=1..2*i_1}...Sum_{i_n=1..2*i_(n-1)} (2*i_n). For example:
a(3) = Sum_{i=1..2} 2*i.
a(4) = Sum_{i=1..2}Sum_{j=1..2*i} 2*j.
a(5) = Sum_{i=1..2}Sum_{j=1..2*i}Sum_{k=1..2*j} 2*k. (End)
The conjecture is true: see Links. - Chams Lahlou, Jan 26 2019

Recurrence and more terms from Michael Kleber, Dec 05 2000

A068413 a(n) = number of partitions of 2^n.

Original entry on oeis.org

1, 2, 5, 22, 231, 8349, 1741630, 4351078600, 365749566870782, 4453575699570940947378, 61847822068260244309086870983975, 18116048323611252751541173214616030020513022685, 6927233917602120527467409170319882882996950147283323368445315320451

Offset: 0

Henry Bottomley, Mar 03 2002

nonn

			a(2)=5 since there are 5 partitions of 2^2=4: 4, 3+1, 2+2, 2+1+1, 1+1+1+1+1.

Alois P. Heinz, Table of n, a(n) for n = 0..19
Henry Bottomley, Partition calculators using java applets
Index entries for sequences related to partitions

Cf. A000041, A000079, A018819, A067735.

Mathematica
```
Table[ PartitionsP[2^n], {n, 0, 12}]
```

a(n) = A000041(A000079(n)).

a(n) ~ exp(Pi*sqrt(2^(n+1)/3))/(sqrt(3)*2^(n+2)). - Ilya Gutkovskiy, Jan 13 2017

A318400 Numbers whose prime indices are all powers of 2 (including 1).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 19, 21, 24, 27, 28, 32, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 64, 72, 76, 81, 84, 96, 98, 106, 108, 112, 114, 126, 128, 131, 133, 144, 147, 152, 159, 162, 168, 171, 189, 192, 196, 212, 216, 224, 228, 243, 252, 256, 262

Offset: 1

Gus Wiseman, Dec 16 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of all integer partitions whose parts are all powers of 2 (including 1) begins: (), (1), (2), (11), (21), (4), (111), (22), (211), (41), (1111), (221), (8), (42), (2111), (222), (411), (11111), (2211), (81), (421), (21111), (44).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003963, A018819, A033844, A056239, A106349, A112798, A302242, A302491, A322551.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pow2Q[n_]:=Or[n==1,MatchQ[FactorInteger[n],{{2,_}}]];
Select[Range[100],And@@pow2Q/@primeMS[#]&]

Sum_{n>=1} 1/a(n) = 1/Product_{k>=0} (1 - 1/prime(2^k)) = 3.81625872357742992578... . - Amiram Eldar, Dec 03 2022

A323093 Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 9, 12, 13, 18, 23, 29, 37, 49, 55, 71, 84, 104, 126, 153, 185, 221, 261, 317, 375, 446, 523, 623, 721, 854, 994, 1168, 1357, 1579, 1833, 2126, 2455, 2843, 3270, 3766, 4320, 4980, 5687, 6521, 7444, 8498, 9684, 11039, 12540, 14262

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			The a(1) = 1 through a(8) = 12 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (311)    (51)      (52)       (53)
                    (1111)  (11111)  (222)     (61)       (62)
                                     (3111)    (322)      (71)
                                     (111111)  (331)      (332)
                                               (511)      (611)
                                               (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (311111)
                                                          (11111111)

Cf. A018819, A101417, A120641, A276431, A305148, A323092, A323094.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[#,IntegerQ[Log[2,#1/#2]]&]&]],{n,30}]

A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 7, 7, 10, 9, 15, 13, 21, 19, 28, 26, 40, 35, 54, 49, 72, 64, 97, 87, 128, 115, 167, 151, 220, 195, 284, 256, 366, 328, 469, 421, 598, 537, 757, 682, 959, 859, 1204, 1085, 1507, 1354, 1880, 1694, 2338, 2104, 2892, 2609, 3574, 3218, 4394

Offset: 0

Gus Wiseman, Jan 31 2022

nonn

These are partitions of n with all even multiplicities (or run-lengths), except possibly the last.

			The a(1) = 1 through a(9) = 7 partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    221    33      331      44        333
              1111  11111  222     22111    332       441
                           2211    1111111  2222      22221
                           111111           3311      33111
                                            221111    2211111
                                            11111111  111111111

The ordered version (compositions) is A016116.
The even-length case is A035363.
A reverse version is A096441, both A349060.
The version for unequal instead of equal is A122129, even-length A351008.
The version for even instead of odd indices is A351003, even-length A351012.
The strict version is A351005, opposite A351006, even-length A035457.
Cf. A000070, A018819, A027383, A088218, A067661, A101417, A122134, A122135, A350842, A351007.

Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A089052 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of n into exactly k powers of 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 1, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1

Offset: 0

N. J. A. Sloane, Dec 03 2003

			1
0 1
0 1 1
0 0 1 1
0 1 1 1 1
0 0 1 1 1 1
0 0 1 2 1 1 1
0 0 0 1 2 1 1 1
0 1 1 1 2 2 1 1 1
0 0 1 1 1 2 2 1 1 1
0 0 1 2 2 2 2 2 1 1 1
0 0 0 1 2 2 2 2 2 1 1 1

J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From N. J. A. Sloane, Feb 03 2013

Alois P. Heinz, Rows n = 0..200, flattened
J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From _N. J. A. Sloane_, Feb 03 2013

Columns give A036987, A075897 (essentially), A089049, A089050, A089051, A319922.
Row sums give A018819.
See A089053 for another version.

A089052 := proc(n, k)
    option remember;
    if k > n then
        return(0);
    end if;
    if k= 0 then
        if n=0 then
            return(1)
        else
            return(0);
        end if;
    end if;
    if n mod 2 = 1 then
            return procname(n-1, k-1);
    end if;
    procname(n-1, k-1)+procname(n/2, k);
end proc:

t[n_, k_] := t[n, k] = Which[k > n, 0, k == 0, If[n == 0, 1, 0], Mod[n, 2] == 1, t[n-1, k-1], True, t[n-1, k-1] + t[n/2, k]]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Maple *)

T(2m, k) = T(m, k)+T(2m-1, k-1); T(2m+1, k) = T(2m, k-1).

G.f.: 1/Product_{k>=0} (1-y*x^(2^k)). - Vladeta Jovovic, Dec 03 2003

A171238 Given M = triangle A122196 as an infinite lower triangular matrix, this sequence is lim_{k->infinity} M^k.

Original entry on oeis.org

1, 2, 5, 8, 16, 24, 40, 56, 88, 120, 176, 232, 328, 424, 576, 728, 968, 1208, 1568, 1928, 2464, 3000, 3768, 4536, 5632, 6728, 8248, 9768, 11864, 13960, 16784, 19608, 23400, 27192, 32192, 37192, 43760, 50328, 58824, 67320, 78280, 89240, 103200, 117160

Offset: 1

Gary W. Adamson, Dec 05 2009

nonn

Also equals polcoeff: (1,2,3,...)*(1,0,2,0,5,0,8,0,16,...).
Number of binary partitions of n into two kinds of parts. - Joerg Arndt, Feb 26 2015
Let the n-th convolution power of the sequence = B, with C = the aerated variant of B. It appears that B/C = the binomial sequence starting (1, 2n, ...). Example: The sequence squared = (1, 4, 14, 36, 89, 192, ...) = B; with C = (1, 0, 4, 0, 14, 0, 36, ...). Then B/C = A000292: (1, 4, 10, 20, 35, 56, ...). - Gary W. Adamson, Aug 15 2016

			G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 16*x^5 + 24*x^6 + 40*x^7 + 56*x^8 + ...

Georg Fischer, Table of n, a(n) for n = 1..1000 [first 128 terms from Vincenzo Librandi]

Cf. A000292, A018819, A122196.

imax=10; CoefficientList[ Series[ 1/ Product[1 - x^(2^i), {i, 0, imax}]^2, {x, 0, 2^imax}], x] (* Robert G. Wilson v, May 11 2012; range of "i" amended by Georg Fischer, May 12 2024 *)

Given M = triangle A122196 as an infinite lower triangular matrix, this sequence is lim_{k->infinity}, a left-shifted vector considered as a sequence.
From Wolfdieter Lang, Jul 15 2010: (Start)
O.g.f.: x*Q(x) with Q(x)*(1-x)^2 = Q(x^2), for the eigensequence M*Q = Q with the column o.g.f.s (x^(2*m))/(1-x)^2, m >= 0, of M.
Recurrence for b(n):=a(n+1): b(n)=0 if n < 0, b(0)=1; if n is even then b(n) = b(n/2) + 2*b(n-1) - b(n-2), otherwise b(n) = 2*b(n-1) - b(n-2). (End)
G.f.: 1/((1-x)*(1-x^2)*(1-x^4)* ... *(1- x^(2^k))* ...)^2. - Robert G. Wilson v, May 11 2012
Convolution square of A018819. - Michael Somos, Mar 28 2014

More terms from Wolfdieter Lang, Jul 15 2010

cp's OEIS Frontend

A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

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A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

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A350844 Number of strict integer partitions of n with no difference -2.

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A002449 Number of different types of binary trees of height n.

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PARI

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A068413 a(n) = number of partitions of 2^n.

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A318400 Numbers whose prime indices are all powers of 2 (including 1).

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A323093 Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.

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A351004 Alternately constant partitions. Number of integer partitions y of n such that y_i = y_{i+1} for all odd i.