A000123 -id:A000123 - cp's OEIS frontend

A089177 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n))) giving number of non-squashing partitions of n into k parts.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 4, 4, 1, 1, 5, 6, 2, 1, 6, 9, 4, 1, 7, 12, 6, 1, 8, 16, 10, 1, 1, 9, 20, 14, 2, 1, 10, 25, 20, 4, 1, 11, 30, 26, 6, 1, 12, 36, 35, 10, 1, 13, 42, 44, 14, 1, 14, 49, 56, 20, 1, 15, 56, 68, 26, 1, 16, 64, 84, 36, 1, 1, 17, 72, 100, 46, 2, 1, 18, 81, 120, 60, 4, 1

Offset: 0

N. J. A. Sloane, Dec 08 2003

T(n,k) = A181322(n,k) - A181322(n,k-1) for n>0. - Alois P. Heinz, Jan 25 2014

			Triangle begins:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  2;
  1, 4,  4,  1;
  1, 5,  6,  2;
  1, 6,  9,  4;
  1, 7, 12,  6;
  1, 8, 16, 10,  1;

Alois P. Heinz, Rows n = 0..1002, flattened
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003; Discrete Math., 294 (2005), 259-274.

Cf. A078121, A089178. Columns give A002620, A008804, A088932, A088954. Row sums give A000123.

T:= proc(n) option remember;
     `if`(n=0, 1, zip((x, y)-> x+y, [T(n-1)], [0, T(floor(n/2))], 0)[])
    end:
seq(T(n), n=0..25);  # Alois P. Heinz, Apr 01 2012

row[0] = {1}; row[1] = {1, 1}; row[n_] := row[n] = Plus @@ PadRight[ {row[n-1], Join[{0}, row[Floor[n/2]]]} ]; Table[row[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Jan 31 2014 *)

Row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor(n/2) shifted one place right).
G.f. for column k (k >= 2): x^(2^(k-2))/((1-x)*Product_{j=0..k-2} (1-x^(2^j))). [corrected by Jason Yuen, Jan 12 2025]
Conjecture: let R(n,x) be the n-th reversed row polynomial, then R(n,x) = Sum_{k=0..A000523(A053645(n)) + 1} T(A053645(n),k)*R(2^(A000523(n)-k),x) for n > 0, n != 2^m with R(0,x) = 1 where R(2^m,x) is the (m+1)-th row polynomial of A078121. - Mikhail Kurkov, Jun 28 2025

More terms from Alford Arnold, May 22 2004

A089292 G.f.: Product_{m>=1} 1/(1-x^m)^A018819(m).

Original entry on oeis.org

1, 1, 3, 5, 12, 20, 41, 69, 132, 222, 399, 665, 1156, 1904, 3212, 5234, 8645, 13925, 22596, 36008, 57590, 90862, 143508, 224316, 350505, 543159, 840623, 1292317, 1983094, 3026178, 4608061, 6983663, 10559800, 15901698, 23889722, 35760786, 53405395, 79498207

Offset: 0

N. J. A. Sloane, Dec 24 2003

nonn

Number of 2-dimensional partitions of n where each row is non-squashing.

			a(4) = 12:
4.31.3.22.2.211.21.2..2.11.11.1
.....1....2.....1..11.1.11.1..1
......................1....1..1
..............................1
211 and 1111 for example are excluded because they would squash.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Cf. A000123, A018819, A001970.

maxm = 38;
b[0] = b[1] = 1; b[n_] := b[n] = If[OddQ[n], b[n-1], b[n-1] + b[n/2]];
Product[1/(1-x^m)^b[m], {m, 1, maxm}] + O[x]^maxm // CoefficientList[#, x]&
(* Jean-François Alcover, Oct 02 2018 *)

A131205 a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)).

Original entry on oeis.org

1, 3, 7, 13, 23, 37, 57, 83, 119, 165, 225, 299, 393, 507, 647, 813, 1015, 1253, 1537, 1867, 2257, 2707, 3231, 3829, 4521, 5307, 6207, 7221, 8375, 9669, 11129, 12755, 14583, 16613, 18881, 21387, 24177, 27251, 30655, 34389, 38513, 43027, 47991

Offset: 1

Reinhard Zumkeller, Oct 22 2007

nonn

From Gary W. Adamson, Dec 16 2009: (Start)
Let M = an infinite lower triangular matrix with (1, 3, 4, 4, 4, ...) in every column shifted down twice, with the rest zeros:
  1;
  3, 0;
  4, 1, 0;
  4, 3, 0, 0;
  4, 4, 1, 0, 0;
  4, 4, 3, 0, 0, 0;
  ...
A131205 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. (End)
The subsequence of primes in this sequence begins with 5 in a row: 3, 7, 13, 23, 37, 83, 647, 1867, 2707, 88873, 388837, 655121, 754903, 928621, 1062443. - Jonathan Vos Post, Apr 25 2010

T. D. Noe, Table of n, a(n) for n = 1..1000
Cristina Ballantine, George Beck, and Mircea Merca, Partitions and elementary symmetric polynomials -- an experimental approach, arXiv:2408.13346 [math.CO], 2024. See p. 13.
Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See pp. 5, 7.

Cf. A000123, A008619. Bisection of A033485.

a131205 n = a131205_list !! (n-1)
a131205_list = scanl1 (+) a000123_list -- Reinhard Zumkeller, Oct 10 2013

A[1]:= 1:
for n from 2 to 100 do A[n]:= A[n-1] + A[floor(n/2)] + A[ceil(n/2)] od:
seq(A[n],n=1..100); # Robert Israel, Sep 06 2016

Nest[Append[#1, #1[[-1]] + #1[[Floor@ #3]] + #[[Ceiling@ #3]] ] & @@ {#1, #2, #2/2} & @@ {#, Length@ # + 1} &, {1}, 42] (* Michael De Vlieger, Jan 16 2020 *)

Partial sums of A000123. - Gary W. Adamson, Oct 26 2007
G.f.: r(x) * r(x^2) * r(x^4) * r(x^8) * ... where r(x) = (1 + 3x + 4x^2 + 4x^3 + 4x^4 + ...) is the g.f. of A113311. - Gary W. Adamson, Sep 01 2016
G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k))/(1 - x^(2^k)). - Ilya Gutkovskiy, Jun 05 2017
a(n) = A033485(2n-1). - Jean-Paul Allouche, Aug 11 2021

A258485 Number of tangled chains of length k=7.

Original entry on oeis.org

1, 1, 365, 7119961, 1172597933594, 934741501255380321, 2602204282373953017437500, 20410544568790568555722851029455, 387481340785957748099474582410763014214, 15899856312608503503306403988460714538830399657

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=6, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^7)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

A292477 Square array A(n,k), n >= 0, k >= 2, read by antidiagonals: A(n,k) = [x^(k*n)] Product_{j>=0} 1/(1 - x^(k^j)).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 6, 1, 2, 3, 5, 10, 1, 2, 3, 4, 7, 14, 1, 2, 3, 4, 6, 9, 20, 1, 2, 3, 4, 5, 8, 12, 26, 1, 2, 3, 4, 5, 7, 10, 15, 36, 1, 2, 3, 4, 5, 6, 9, 12, 18, 46, 1, 2, 3, 4, 5, 6, 8, 11, 15, 23, 60, 1, 2, 3, 4, 5, 6, 7, 10, 13, 18, 28, 74, 1, 2, 3, 4, 5, 6, 7, 9, 12, 15, 21, 33, 94

Offset: 0

Ilya Gutkovskiy, Sep 17 2017

A(n,k) is the number of partitions of k*n into powers of k.

			Square array begins:
   1,  1,  1,  1,  1,  1, ...
   2,  2,  2,  2,  2,  2, ...
   4,  3,  3,  3,  3,  3, ...
   6,  5,  4,  4,  4,  4, ...
  10,  7,  6,  5,  5,  5, ...
  14,  9,  8,  7,  6,  6, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
George E. Andrews, Aviezri S. Fraenkel, James A. Sellers, Characterizing the number of m-ary partitions modulo m, Amer. Math. Monthly 122:9 (2015), 880-885.
Index entries for related partition-counting sequences

Columns k=2..5 give A000123, A005704, A005705, A005706.
Mirror of A089688 (excluding the first row).
Cf. A145515, A294316.

Table[Function[k, SeriesCoefficient[Product[1/(1 - x^k^i), {i, 0, n}], {x, 0, k n}]][j - n + 2], {j, 0, 12}, {n, 0, j}] // Flatten

A322156 Irregular triangle where row n includes all decreasing sequences S = {k_0 = n, k_1, k_2, ..., k_m} in reverse lexicographic order such that the sum of subsequent terms k_j for all i < j <= m does not exceed any k_i.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 3, 3, 1, 3, 1, 1, 3, 2, 3, 2, 1, 3, 3, 4, 4, 1, 4, 1, 1, 4, 2, 4, 2, 1, 4, 2, 1, 1, 4, 2, 2, 4, 3, 4, 3, 1, 4, 4, 5, 5, 1, 5, 1, 1, 5, 2, 5, 2, 1, 5, 2, 1, 1, 5, 2, 2, 5, 3, 5, 3, 1, 5, 3, 1, 1, 5, 3, 2, 5, 4, 5, 4, 1, 5, 5, 6, 6, 1, 6, 1, 1, 6, 2, 6, 2, 1, 6, 2, 1, 1, 6, 2, 2, 6

Offset: 1

Michael De Vlieger, Dec 11 2018

Algorithm:
Let S be a sequence starting with n. Let k be the index of a term in S, with n at position k = 0. Let S_r be the r-th sequence in row n.
Starting with S_1 = {n}, we either (A) append a 1 to the left of S_r, or (B) we drop the most recently-appended term S_(k) and increment the rightmost term (k - 1).
By default we execute (A) and test according to the following. Consider the reversed accumulation A_(r + 1) = Sum(reverse(S_(k + 1))) = Sum(k_m, k_(m - 1), ..., k_2, k_1). If S_r - A_(r + 1) contains nothing less than 0, then S_(k + 1) is retained, otherwise we execute (B).
We end after k_1 = n, since otherwise we would enter an endless loop that also increments k_0 ad infinitum.
The first sequence S in row n is {n} while the last is {n, n}.
All rows n contain {{n}, {n, 1}, {n, n}}.
Only one repeated term k may appear at the end of any S in row n.
The longest possible sequence S in row n has 2 + floor(log_2(n)) terms = 2 + A113473(n).
The sequence S describes unique integer partitions L that are recursively symmetrical. Example: We can convert S = {4, 2, 1} into the partition (7, 6, 5, 4, 3, 2, 1), a partition of N = 28. We set a 4X Durfee square with its upper-left corner at origin. Then we set 2^k = 2^1 = 2 2X squares, each with its upper-left corner in any coordinate bounded at left and top by either a previously-lain square or an axis. Finally, we set 2^2 = 4 1X squares as above once again. We obtain a Ferrer diagram as below, with the k marked, i.e., the 1st term 4X, the 2nd term 2X, the 3rd term 1X squares:
    0  0  0  0  1  1  2
    0  0  0  0  1  1
    0  0  0  0  2
    0  0  0  0
    1  1  2
    1  1
    2
The resulting partition L is recursively self-conjugate; its arms are identical to its legs. We can eliminate the Durfee square and the other appendage and have a symmetrical partition L_1 with Durfee square of k_1 units, etc.
Were we to admit either more than 1 repeated k or a term such that S_k - A_(k + 1) had differences less than 1, we would have overlapping squares in the Ferrer diagram. Such diagrams are generated by larger n and all resulting diagrams are unique given the described algorithm.
The sequences S in row n, converted into integer partitions L, sum to n^2 <= N <= 3 * n^2.

			Triangle begins:
1; 1,1;
2; 2,1; 2,1,1; 2,2;
3; 3,1; 3,1,1; 3,2; 3,2,1; 3,3;
4; 4,1; 4,1,1; 4,2; 4,2,1; 4,2,1,1; 4,2,2; 4,3; 4,3,1; 4,4;
...
Row n = 5 starts with S_1 = 5. We append 1 to get {5,1}. 1 does not exceed 5, thus S_2 = {5,1}. We append 1 to get {5,1,1}. A = {1,2}; {5,1}-{2,1} = {3,0}, thus S_3 = {5,1,1} and we drop the last term and increment the new last term to get {5,2}. S_4 = {5,2}, and the ensuing terms {5,2,1}, {5,2,1,1}, {5,2,2} enter into the row. Since there are repeated terms at the last sequence, we drop the last term and increment the new last to get {5,3}. The terms {5,3,1}, {5,3,1,1}, {5,3,2}, {5,3,2,1}, are admitted. {5,3,2,1,1} has A = {1,2,4,6}. {5,3,2,1}-{6,4,2,1} = {-1,1,0,0}: {5,3,2,1,1} cannot be admitted, so we drop the last term and increment to {5,3,2,2} but the sum of the last two terms exceeds the second and we drop the last term and increment to {5,3,3}. For similar reasons, this cannot be admitted, so we drop the last term and increment to {5,4}. This enters as well as {5,4,1}. Since any appendage or increment proves invalid, we end up incrementing to {5,5}. The two terms are the same, therefore we end the row n = 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10055 (rows 1 <= n <= 21, flattened)
Michael De Vlieger, Illustration for A322156
Michael De Vlieger, Rows 1 <= n <= 30, sequences S in row n have terms k that are space delimited

Cf. A000123, A033485, A190899, A190900, A321223, A322457.

(* Generate sequence: *)
f[n_] := Block[{w = {n}, c}, c[x_] := Apply[Times, Most@ x - Reverse@ Accumulate@ Reverse@ Rest@ x]; Reap[Do[Which[And[Length@ w == 2, SameQ @@ w], Sow[w]; Break[], Length@ w == 1, Sow[w]; AppendTo[w, 1], c[w] > 0, Sow[w]; AppendTo[w, 1], True, Sow[w]; w = MapAt[1 + # &, Drop[w, -1], -1]], {i, Infinity}] ][[-1, 1]] ]; Array[f, 6] // Flatten
(* Convert S = row n to standard partition: *)
g[w_] := Block[{k}, k = Total@ w; Total@ Map[Apply[Function[{s, t}, s Array[Boole[t <= # <= s + t - 1] &, k] ], #] &, Apply[Join, Prepend[Table[Function[{v, c}, Map[{w[[k]], # + 1} &, Map[Total[v #] &, Tuples[{0, 1}, {Length@ v}]]]] @@ {Most@ #, ConstantArray[1, Length@ # - 1]} &@ Take[w, k], {k, 2, Length@ w}], {{w[[1]], 1}}]]] ]

Row n contains A000123(n) = 2*A033485(n) sequences S.

A088954 G.f.: 1/((1-x)^2(1-x^2)(1-x^4)(1-x^8)(1-x^16)).

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, 94, 114, 140, 166, 202, 238, 284, 330, 390, 450, 524, 598, 692, 786, 900, 1014, 1154, 1294, 1460, 1626, 1827, 2028, 2264, 2500, 2780, 3060, 3384, 3708, 4088, 4468, 4904, 5340, 5844, 6348, 6920, 7492, 8148, 8804, 9544

Offset: 0

N. J. A. Sloane, Dec 02 2003

nonn

a(n) is the number of partitions of 2*n into powers of 2 less than or equal to 2^5. First differs from A000123 at n=32. - Alois P. Heinz, Apr 02 2012

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 2, -2, 0, 2, 0, -2, 0, 2, -2, 2, 0, -2, 2, -2, 0, 2, -2, 2, 0, -2, 0, 2, 0, -2, 2, -2, 0, 2, -1).

See A000027, A002620, A008804, A088932, A000123 for similar sequences.

Column k=5 of A181322.

f := proc(n,k) option remember; if k > n then RETURN(0); fi; if k= 0 then if n=0 then RETURN(1) else RETURN(0); fi; fi; if k = 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(f(n-1,k)); fi; f(n-1,k)+f(n/2,k-1); end; # present sequence is f(2m,6)
GFF := k->x^(2^(k-2))/((1-x)*mul((1-x^(2^j)),j=0..k-2)); # present g.f. is GFF(6)/x^16
a:= proc(n) local m, r; m:= iquo(n, 16, 'r'); r:= r+1; [1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, 94, 114, 140, 166][r] +(((((128/5*m +8*(15+r))*m +(228 +[0, 32, 68, 104, 144, 184, 228, 272, 320, 368, 420, 472, 528, 584, 644, 704][r]))*m +(172 +[0, 43, 98, 153, 223, 293, 378, 463, 566, 669, 790, 911, 1053, 1195, 1358, 1521][r]))*m +(247/5 +[0, 22, 55, 88, 138, 188, 255, 322, 415, 508, 627, 746, 900, 1054, 1243, 1432][r]))*m)/3 end: seq(a(n), n=0..60); # Alois P. Heinz, Apr 17 2009

CoefficientList[Series[1/((1-x)^2(1-x^2)(1-x^4)(1-x^8)(1-x^16)),{x,0,70}],x] (* or *) LinearRecurrence[{2,0,-2,2,-2,0,2,0,-2,0,2,-2,2,0,-2,2,-2,0,2,-2,2,0,-2,0,2,0,-2,2,-2,0,2,-1},{1,2,4,6,10,14,20,26,36,46,60,74,94,114,140,166,202,238,284,330,390,450,524,598,692,786,900,1014,1154,1294,1460,1626},70](* Harvey P. Dale, Feb 12 2013 *)

a(0)=1, a(1)=2, a(2)=4, a(3)=6, a(4)=10, a(5)=14, a(6)=20, a(7)=26, a(8)=36, a(9)=46, a(10)=60, a(11)=74, a(12)=94, a(13)=114, a(14)=140, a(15)=166, a(16)=202, a(17)=238, a(18)=284, a(19)=330, a(20)=390, a(21)=450, a(22)=524, a(23)=598, a(24)=692, a(25)=786, a(26)=900, a(27)=1014, a(28)=1154, a(29)=1294, a(30)=1460, a(31)=1626, a(n)=2*a(n-1)-2*a(n-3)+ 2*a(n-4)- 2*a(n-5)+ 2*a(n-7)-2*a(n-9)+2*a(n-11)-2*a(n-12)+2*a(n-13)-2*a(n-15)+2*a(n-16)-2*a(n-17)+ 2*a(n-19)- 2*a(n-20)+ 2*a(n-21)-2*a(n-23)+2*a(n-25)-2*a(n-27)+2*a(n-28)-2*a(n-29)+ 2*a(n-31)-a(n-32). - Harvey P. Dale, Feb 12 2013

A100529 a(n) = minimal k such that n has a partition into k parts with the property that every number <= m can be partitioned into a subset of these parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 3, 4, 2, 2, 1, 1, 12, 15, 13, 14, 11, 12, 9, 10, 6, 6, 4, 4, 2, 2, 1, 1, 84, 91, 82, 89, 77, 80, 70, 73, 60, 63, 53, 54, 43, 44, 35, 36, 26, 26, 20, 20, 14, 14, 10, 10, 6, 6, 4, 4, 2, 2, 1, 1, 908

Offset: 1

N. J. A. Sloane, Dec 31 2004

nonn

E. O'Shea, M-partitions: optimal partitions of weight for one scale pan, Discrete Math. 289 (2004), 81-93.
O. J. Rodseth, Enumeration of M-partitions, Discrete Math., 306 (2006), 694-698.

Cf. A000123 (binary partitions), A002033 (perfect partitions).

If 2^m + 2^(m-1) - 1 <= n <= 2^(m+1) - 1 for some m, let i = 2^(m+1) - 1 - n. Then a(n) = A000123([i/2]). This determines half the values.

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)}

cp's OEIS Frontend

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