A066208 -id:A066208 - cp's OEIS frontend

A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

1, 1, 0, 2, 1, 0, 0, 3, 0, 2, 1, 0, 0, 0, 0, 5, 1, 0, 0, 4, 0, 2, 1, 0, 2, 0, 0, 0, 0, 0, 1, 7, 0, 2, 0, 0, 0, 0, 0, 7, 1, 0, 0, 4, 0, 2, 1, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 11, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 12, 0, 2, 1, 0, 2, 0

Offset: 1

Gus Wiseman, Sep 08 2022

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 a(440) = 21 multiset partitions of {1,1,1,3,5}:
  {1}{1}{1}{3}{5}  {1}{1}{1}{35}  {1}{1}{135}  {1}{1135}  {11135}
                   {1}{1}{13}{5}  {1}{11}{35}  {11}{135}
                   {1}{11}{3}{5}  {11}{13}{5}  {111}{35}
                   {1}{1}{3}{15}  {1}{13}{15}  {113}{15}
                                  {11}{3}{15}  {13}{115}
                                  {1}{3}{115}  {3}{1115}
                                  {1}{5}{113}  {5}{1113}
                                  {3}{111}{5}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Positions of 0's are A324929, complement A066208.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A356069 counts gapless divisors, initial A356224 (complement A356225).
Other types: A000009, A000012, A000041, A089259, A356930.
Other conditions: A050320, A050330, A356936, A322585, A356233, A356945.
Cf. A003963, A073491, A107742, A287170, A325534, A325535, A340852, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],And@@(OddQ[Times@@primeMS[#]]&/@#)&]],{n,100}]

a(n) = 0 if n is in A324929, otherwise a(n) = A001055(n).

A360646 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A066208(n) * A066207(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 12, 14, 9, 8, 15, 28, 18, 13, 10, 24, 35, 36, 26, 19, 11, 30, 56, 45, 52, 38, 21, 16, 33, 70, 72, 65, 76, 42, 27, 17, 48, 77, 90, 104, 95, 84, 54, 29, 20, 51, 112, 99, 130, 152, 105, 108, 58, 37, 22, 60, 119, 144, 143, 190, 168, 135, 116, 74, 39

Offset: 1

Rémy Sigrist, Feb 15 2023

Every positive integer occurs exactly once.

			Array A(n, k) begins:
  n\k |  1   2    3    4    5    6    7    8    9   10
  ----+-----------------------------------------------
    1 |  1   3    7    9   13   19   21   27   29   37
    2 |  2   6   14   18   26   38   42   54   58   74
    3 |  4  12   28   36   52   76   84  108  116  148
    4 |  5  15   35   45   65   95  105  135  145  185
    5 |  8  24   56   72  104  152  168  216  232  296
    6 | 10  30   70   90  130  190  210  270  290  370
    7 | 11  33   77   99  143  209  231  297  319  407
    8 | 16  48  112  144  208  304  336  432  464  592
    9 | 17  51  119  153  221  323  357  459  493  629
   10 | 20  60  140  180  260  380  420  540  580  740

Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A249327, A066207, A066208, A247503, A248101.

PARI
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See Links section.
```

A(n, 1) = A066208(n) = A247503(A(n,k)).

A(1, k) = A066207(k) = A248101(A(n,k)).

A374611 If n = A066207(i) * A066208(j) for some i, j > 0 then a(n) = A066207(j) * A066208(i).

Original entry on oeis.org

1, 3, 2, 7, 9, 6, 4, 13, 5, 19, 21, 14, 8, 12, 18, 27, 29, 15, 10, 37, 11, 39, 43, 26, 49, 24, 16, 28, 17, 38, 53, 57, 42, 61, 36, 35, 20, 30, 22, 63, 71, 33, 23, 79, 45, 81, 87, 54, 25, 89, 58, 56, 31, 48, 91, 52, 32, 51, 101, 74, 34, 107, 40, 111, 72, 78

Offset: 1

Rémy Sigrist, Jul 14 2024

nonn

This sequence is a self-inverse permutation of the positive integers with infinitely many fixed points.

			For n = 42: 42 = 21 * 2 = A066207(7) * A066208(2), so a(42) = A066207(2) * A066208(7) = 3 * 11 = 33.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

See A374600 for a similar sequence.

Cf. A066207, A066208, A247503, A248101, A360646.

PARI
```
\\ See Links section.
```

a(A360646(n, k)) = A360646(k, n).

a(n) = n iff n = A066207(k) * A066208(k) for some k > 0.

A027193 Number of partitions of n into an odd number of parts.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 29, 37, 52, 66, 90, 113, 151, 190, 248, 310, 400, 497, 632, 782, 985, 1212, 1512, 1851, 2291, 2793, 3431, 4163, 5084, 6142, 7456, 8972, 10836, 12989, 15613, 18646, 22316, 26561, 31659, 37556, 44601, 52743, 62416, 73593, 86809, 102064, 120025, 140736

Offset: 0

Clark Kimberling

nonn

Number of partitions of n in which greatest part is odd.
Number of partitions of n+1 into an even number of parts, the least being 1. Example: a(5)=4 because we have [5,1], [3,1,1,1], [2,1,1] and [1,1,1,1,1,1].
Also number of partitions of n+1 such that the largest part is even and occurs only once. Example: a(5)=4 because we have [6], [4,2], [4,1,1] and [2,1,1,1,1]. - Emeric Deutsch, Apr 05 2006
Also the number of partitions of n such that the number of odd parts and the number of even parts have opposite parities.  Example: a(8)=10 is a count of these partitions: 8, 611, 521, 431, 422, 41111, 332, 32111, 22211, 2111111. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
In Chaves 2011 see page 38 equation (3.20). - Michael Somos, Dec 28 2014
Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 10*x^8 + 16*x^9 + 20*x^10 + ...
From _Gus Wiseman_, Feb 11 2021: (Start)
The a(1) = 1 through a(8) = 10 partitions into an odd number of parts are the following. The Heinz numbers of these partitions are given by A026424.
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)
            (111)  (211)  (221)    (222)    (322)      (332)
                          (311)    (321)    (331)      (422)
                          (11111)  (411)    (421)      (431)
                                   (21111)  (511)      (521)
                                            (22111)    (611)
                                            (31111)    (22211)
                                            (1111111)  (32111)
                                                       (41111)
                                                       (2111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part is odd are the following. The Heinz numbers of these partitions are given by A244991.
  (1)  (11)  (3)    (31)    (5)      (33)      (7)        (53)
             (111)  (1111)  (32)     (51)      (52)       (71)
                            (311)    (321)     (322)      (332)
                            (11111)  (3111)    (331)      (521)
                                     (111111)  (511)      (3221)
                                               (3211)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (32111)
                                                          (311111)
                                                          (11111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 39, Example 7.

T. D. Noe, Table of n, a(n) for n = 0..999
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
D. R. C. Chaves, Um estudo combinatório e comparativo de identidades teta parciais de Andrews e Ramanujan, 2011. In Portuguese.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_0(n).

Cf. A000041, A000700, A000701, A026837, A046682.
The Heinz numbers of these partitions are A026424 or A244991.
The even-length version is A027187.
The case of odd sum as well as length is A160786, ranked by A340931.
The case of odd maximum as well as length is A340385.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A067659 counts strict partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
A000009 counts partitions into odd parts, ranked by A066208.
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A101707 counts partitions of odd positive rank.
Cf. A078408, A174725, A236914, A300272, A340831.

g:=sum(x^(2*k)/product(1-x^j,j=1..2*k-1),k=1..40): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..45); # Emeric Deutsch, Apr 05 2006

nn=40;CoefficientList[Series[ Sum[x^(2j+1)Product[1/(1- x^i),{i,1,2j+1}],{j,0,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Dec 01 2012 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n], OddQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n], OddQ[ First@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n + 1], #[[-1]] == 1 && EvenQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n + 1], EvenQ[ First@#] && (Length[#] < 2 || #[[1]] != #[[2]]) &]]; (* Michael Somos, Dec 28 2014 *)

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, if( k%2, x^k / prod( j=1, k, 1 - x^j, 1 + x * O(x^(n-k)) ))), n))}; /* Michael Somos, Jul 24 2012 */

q='q+O('q^66); concat([0], Vec( (1/eta(q)-eta(q)/eta(q^2))/2 ) ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) - (-1)^n*A000700(n)) / 2.
For g.f. see under A027187.
G.f.: Sum(k>=1, x^(2*k-1)/Product(j=1..2*k-1, 1-x^j ) ). - Emeric Deutsch, Apr 05 2006
G.f.: - Sum(k>=1, (-x)^(k^2)) / Product(k>=1, 1-x^k ). - Joerg Arndt, Feb 02 2014
G.f.: Sum(k>=1, x^(k*(2*k-1)) / Product(j=1..2*k, 1-x^j)). - Michael Somos, Dec 28 2014
a(2*n) = A000701(2*n), a(2*n-1) = A046682(2*n-1); a(n) = A000041(n)-A027187(n). - Reinhard Zumkeller, Apr 22 2006

A027187 Number of partitions of n into an even number of parts.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 40, 49, 69, 86, 118, 146, 195, 242, 317, 392, 505, 623, 793, 973, 1224, 1498, 1867, 2274, 2811, 3411, 4186, 5059, 6168, 7427, 9005, 10801, 13026, 15572, 18692, 22267, 26613, 31602, 37619, 44533, 52815, 62338, 73680, 86716, 102162, 119918

Offset: 0

Clark Kimberling

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]
Number of partitions of n+1 into an odd number of parts, the least being 1.
Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

			G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...
From _Gus Wiseman_, Jan 05 2021: (Start)
The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by A028260.
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (2211)    (2221)    (71)
                              (3111)    (3211)    (2222)
                              (111111)  (4111)    (3221)
                                        (211111)  (3311)
                                                  (4211)
                                                  (5111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)
The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by A244990.
  (2)  (21)  (4)    (41)    (6)      (43)      (8)
             (22)   (221)   (42)     (61)      (44)
             (211)  (2111)  (222)    (421)     (62)
                            (411)    (2221)    (422)
                            (2211)   (4111)    (431)
                            (21111)  (22111)   (611)
                                     (211111)  (2222)
                                               (4211)
                                               (22211)
                                               (41111)
                                               (221111)
                                               (2111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
George E. Andrews and David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 13.
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000701, A046682, A026838, A102462.
The Heinz numbers of these partitions are A028260.
The odd version is A027193.
The strict case is A067661.
The case of even sum as well as length is A236913 (the even bisection).
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A000009 counts partitions into odd parts, ranked by A066208.
A026805 counts partitions whose least part is even.
A072233 counts partitions by sum and length.
A101708 counts partitions of even positive rank.
Cf. A000700, A026424, A058696, A096373, A244990, A300061.

f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* Clark Kimberling, Mar 13 2012 *)
a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* Michael Somos, May 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* Michael Somos, Aug 19 2006 */

my(q='q+O('q^66)); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) + (-1)^n * A000700(n))/2.
a(n) = p(n) - p(n-1) + p(n-4) - p(n-9) + ... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan, Mar 14 2004
From Bill Gosper, Jun 25 2005: (Start)
G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3*q^4 + 3*q^5 + 6*q^6 + ...
  = Sum_{n >= 0} q^(2*n)/(q; q)_{2*n}
  = ((Product_{k >= 1} 1/(1-q^k)) + (Product_{k >= 1} 1/(1+q^k)))/2.
Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 5*q^6 + ...
Then B(q) = Sum_{n >= 0} q^(2*n+1)/(q; q){2*n+1} = ((Product{k >= 1} 1/(1-q^k)) - (Product_{k >= 1} 1/(1+q^k)))/2.
Also we have the following identity involving 2 X 2 matrices:
Product_{k >= 1} [ 1/(1-q^(2*k)), q^k/(1-q^(2*k)) ; q^k/(1-q^(2*k)), 1/(1-q^(2*k)) ]
  = [ A(q), B(q) ; B(q), A(q) ]. (End)
a(2*n) = A046682(2*n), a(2*n+1) = A000701(2*n+1); a(n) = A000041(n)-A027193(n). - Reinhard Zumkeller, Apr 22 2006
Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - Michael Somos, Aug 19 2006
G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - Michael Somos, Aug 19 2006
a(n) = A338914(n) + A096373(n). - Gus Wiseman, Jan 06 2021

Offset changed to 0 by Michael Somos, Jul 24 2012

A035363 Number of partitions of n into even parts.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0

Offset: 0

Olivier Gérard

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009
Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009
Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013
Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.
For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:
  1  .  3   .  5    .  7     .  9      .  B       .  D
        21     41      43       63        65         85
               221     61       81        83         A3
                       421      441       A1         C1
                       2221     621       443        643
                                4221      641        661
                                22221     821        841
                                          4421       A21
                                          6221       4441
                                          42221      6421
                                          222221     8221
                                                     44221
                                                     62221
                                                     422221
                                                     2222221
Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:
  (11)  (22)   (33)    (44)     (55)      (66)
        (211)  (321)   (422)    (532)     (633)
               (3111)  (431)    (541)     (642)
                       (4211)   (5221)    (651)
                       (41111)  (5311)    (6222)
                                (52111)   (6321)
                                (511111)  (6411)
                                          (62211)
                                          (63111)
                                          (621111)
                                          (6111111)
Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:
  (2)  (22)  (222)  (2222)  (22222)  (222222)  (2222222)
       (31)  (321)  (3221)  (32221)  (322221)  (3222221)
             (411)  (3311)  (33211)  (332211)  (3322211)
                    (4211)  (42211)  (333111)  (3332111)
                    (5111)  (43111)  (422211)  (4222211)
                            (52111)  (432111)  (4322111)
                            (61111)  (441111)  (4331111)
                                     (522111)  (4421111)
                                     (531111)  (5222111)
                                     (621111)  (5321111)
                                     (711111)  (5411111)
                                               (6221111)
                                               (6311111)
                                               (7211111)
                                               (8111111)
(End)

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02:  [ 1 1 1 1 3 3 1 1 1 1 ]
03:  [ 1 1 1 4 4 1 1 1 ]
04:  [ 1 1 2 3 3 2 1 1 ]
05:  [ 1 1 5 5 1 1 ]
06:  [ 1 2 4 4 2 1 ]
07:  [ 1 6 6 1 ]
08:  [ 2 2 3 3 2 2 ]
09:  [ 2 5 5 2 ]
10:  [ 3 4 4 3 ]
11:  [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03:  [ 1 1 1 1 3 3 1 1 1 1 ]
04:  [ 1 1 1 2 2 2 2 1 1 1 ]
05:  [ 1 1 1 4 4 1 1 1 ]
06:  [ 1 1 2 3 3 2 1 1 ]
07:  [ 1 1 5 5 1 1 ]
08:  [ 1 2 2 2 2 2 2 1 ]
09:  [ 1 2 4 4 2 1 ]
10:  [ 1 3 3 3 3 1 ]
11:  [ 1 6 6 1 ]
12:  [ 2 2 3 3 2 2 ]
13:  [ 2 5 5 2 ]
14:  [ 3 4 4 3 ]
15:  [ 7 7 ]
(End)
a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016
From _Gus Wiseman_, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
  ()  .  (2)  .  (4)   .  (6)    .  (8)     .  (A)      .  (C)
                 (22)     (42)      (44)       (64)        (66)
                          (222)     (62)       (82)        (84)
                                    (422)      (442)       (A2)
                                    (2222)     (622)       (444)
                                               (4222)      (642)
                                               (22222)     (822)
                                                           (4422)
                                                           (6222)
                                                           (42222)
                                                           (222222)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Robert Price, Table of n, a(n) for n = 0..2001

Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z,Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84);  # Alois P. Heinz, Jun 22 2021

nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023

G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018

A066207 All primes that divide n are of the form prime(2k), where prime(k) is k-th prime.

Original entry on oeis.org

1, 3, 7, 9, 13, 19, 21, 27, 29, 37, 39, 43, 49, 53, 57, 61, 63, 71, 79, 81, 87, 89, 91, 101, 107, 111, 113, 117, 129, 131, 133, 139, 147, 151, 159, 163, 169, 171, 173, 181, 183, 189, 193, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 261, 263, 267, 271, 273

Offset: 1

Leroy Quet, Dec 16 2001

nonn

The partitions into even parts, encoded by their Heinz numbers. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1..r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: 63 ( = 3*3*7) is in the sequence because it is the Heinz number of the partition [2, 2, 4]. - Emeric Deutsch, May 19 2015
Numbers divisible only by primes with even indices (A031215). - Michael De Vlieger, Dec 11 2017
Numbers that are not divisible by any prime with an odd index (A031368). - Antti Karttunen, Jul 18 2020
For every positive integer m there exists a unique ordered pair of positive integers (j,k) such that m = a(j)*A066208(k). - Christopher Scussel, Jul 01 2023

			39 is included because 3 * 13 = prime(2) * prime(6) and 2 and 6 are both even.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (original first 1000 terms from Harry J. Smith)

Cf. A031215, A031368, A066208, A215366.
Cf. A297002 (a permutation).
Numbers in the odd bisection of A336321.

Select[Range[273], AllTrue[PrimePi@ FactorInteger[#][[All, 1]], EvenQ] &] (* Michael De Vlieger, Dec 11 2017, range adjusted by Antti Karttunen, Jul 18 2020 *)

{ n=0; for (m=2, 10^9, f=factor(m); b=1; for(i=1, matsize(f)[1], if (primepi(f[i, 1])%2, b=0; break)); if (b, write("b066207.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 06 2010

isA066207(n) = (!#select(p -> (primepi(p)%2), factor(n)[,1])); \\ Antti Karttunen, Jul 18 2020

Offset changed from 0 to 1 by Harry J. Smith, Feb 06 2010
a(53)-a(58) from Harry J. Smith, Feb 06 2010
a(1) = 1 inserted (and the indexing of the rest of terms changed) by Antti Karttunen, Jul 18 2020

A339890 Number of odd-length factorizations of n into factors > 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 6, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 5, 1, 2, 1, 2, 1, 2, 1, 8, 1, 1, 2, 2, 1, 2, 1, 6, 2, 1, 1, 5, 1, 1, 1

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 24, 48, 60, 72, 96, 120:
  24      48          60       72          96          120
  2*2*6   2*3*8       2*5*6    2*4*9       2*6*8       3*5*8
  2*3*4   2*4*6       3*4*5    2*6*6       3*4*8       4*5*6
          3*4*4       2*2*15   3*3*8       4*4*6       2*2*30
          2*2*12      2*3*10   3*4*6       2*2*24      2*3*20
          2*2*2*2*3            2*2*18      2*3*16      2*4*15
                               2*3*12      2*4*12      2*5*12
                               2*2*2*3*3   2*2*2*2*6   2*6*10
                                           2*2*2*3*4   3*4*10
                                                       2*2*2*3*5

Alois P. Heinz, Table of n, a(n) for n = 1..20000

The case of set partitions (or n squarefree) is A024429.
The case of partitions (or prime powers) is A027193.
The ordered version is A174726 (even: A174725).
The remaining (even-length) factorizations are counted by A339846.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A236914, A320732.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n<2, 0, g(n$2, 1)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ@Length[#]&]],{n,100}]

a(n) + A339846(n) = A001055(n).

A195017 If n = Product_{k >= 1} (p_k)^(c_k) where p_k is k-th prime and c_k >= 0 then a(n) = Sum_{k >= 1} c_k*((-1)^(k-1)).

Original entry on oeis.org

0, 1, -1, 2, 1, 0, -1, 3, -2, 2, 1, 1, -1, 0, 0, 4, 1, -1, -1, 3, -2, 2, 1, 2, 2, 0, -3, 1, -1, 1, 1, 5, 0, 2, 0, 0, -1, 0, -2, 4, 1, -1, -1, 3, -1, 2, 1, 3, -2, 3, 0, 1, -1, -2, 2, 2, -2, 0, 1, 2, -1, 2, -3, 6, 0, 1, 1, 3, 0, 1, -1, 1, 1, 0, 1, 1, 0, -1, -1, 5, -4, 2, 1, 0, 2, 0, -2, 4, -1, 0, -2, 3, 0, 2, 0, 4, 1, -1, -1, 4, -1, 1, 1, 2, -1

Offset: 1

Clark Kimberling, Feb 06 2012

Let p(n,x) be the completely additive polynomial-valued function such that p(1,x) = 0 and p(prime(n),x) = x^(n-1), like is defined in A206284 (although here we are not limited to just irreducible polynomials). Then a(n) is the value of the polynomial encoded in such a manner by n, when it is evaluated at x=-1. - The original definition rewritten and clarified by Antti Karttunen, Oct 03 2018
Positions of 0 give the values of n for which the polynomial p(n,x) is divisible by x+1. For related sequences, see the Mathematica section.
Also the number of odd prime indices of n minus the number of even prime indices of n (both counted with multiplicity), where 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. - Gus Wiseman, Oct 24 2023

			The sequence can be read from a list of the polynomials:
  p(n,x)      with x = -1, gives a(n)
------------------------------------------
  p(1,x) = 0           0
  p(2,x) = 1x^0        1
  p(3,x) = x          -1
  p(4,x) = 2x^0        2
  p(5,x) = x^2         1
  p(6,x) = 1+x         0
  p(7,x) = x^3        -1
  p(8,x) = 3x^0        3
  p(9,x) = 2x         -2
  p(10,x) = x^2 + 1    2.
(The list runs through all the polynomials whose coefficients are nonnegative integers.)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206284, A277322, A284010.
For other evaluation functions of such encoded polynomials, see A001222, A048675, A056239, A090880, A248663.
Cf. A006093, A036689, A078437.
Zeros are A325698, distinct A325700.
For sum instead of count we have A366749 = A366531 - A366528.
A000009 counts partitions into odd parts, ranked by A066208.
A035363 counts partitions into even parts, ranked by A066207.
A112798 lists prime indices, reverse A296150, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A019507, A106529, A239241, A239261, A369180.

b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 200;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
== Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
    Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
      Length[f[n]]}]], {n, 1, z}];
p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
Table[p[n, x] /. x -> 0, {n, 1, z/2}]   (* A007814 *)
Table[p[2 n, x] /. x -> 0, {n, 1, z/2}] (* A001511 *)
Table[p[n, x] /. x -> 1, {n, 1, z}]     (* A001222 *)
Table[p[n, x] /. x -> 2, {n, 1, z}]     (* A048675 *)
Table[p[n, x] /. x -> 3, {n, 1, z}]     (* A090880 *)
Table[p[n, x] /. x -> -1, {n, 1, z}]    (* A195017 *)
z = 100; Sum[-(-1)^k IntegerExponent[Range[z], Prime[k]], {k, 1, PrimePi[z]}] (* Friedjof Tellkamp, Aug 05 2024 *)

A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * (-1)^(1+primepi(f[i,1])))); } \\ Antti Karttunen, Oct 03 2018

Totally additive with a(p^e) = e * (-1)^(1+PrimePi(p)), where PrimePi(n) = A000720(n). - Antti Karttunen, Oct 03 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} = (-1)^(primepi(p)+1)/(p-1) = Sum_{k>=1} (-1)^(k+1)/A006093(k) = A078437 +  Sum_{k>=1} (-1)^(k+1)/A036689(k) = 0.6339266524059... . - Amiram Eldar, Sep 29 2023
a(n) = A257991(n) - A257992(n). - Gus Wiseman, Oct 24 2023
a(n) = -Sum_{k=1..pi(n)} (-1)^k * valuation(n, prime(k)). - Friedjof Tellkamp, Aug 05 2024

More terms, name changed and example-section edited by Antti Karttunen, Oct 03 2018

A087897 Number of partitions of n into odd parts greater than 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, 13, 15, 18, 20, 23, 27, 30, 34, 40, 44, 50, 58, 64, 73, 83, 92, 104, 118, 131, 147, 166, 184, 206, 232, 256, 286, 320, 354, 394, 439, 485, 538, 598, 660, 730, 809, 891, 984, 1088, 1196, 1318, 1454, 1596, 1756

Offset: 0

N. J. A. Sloane, Dec 04 2003

Also number of partitions of n into distinct parts which are not powers of 2.
Also number of partitions of n into distinct parts such that the two largest parts differ by 1.
Also number of partitions of n such that the largest part occurs an odd number of times that is at least 3 and every other part occurs an even number of times. Example: a(10) = 2 because we have [2,2,2,1,1,1,1] and [2,2,2,2,2]. - Emeric Deutsch, Mar 30 2006
Also difference between number of partitions of 1+n into distinct parts and number of partitions of n into distinct parts. - Philippe LALLOUET, May 08 2007
In the Berndt reference replace {a -> -x, q -> x} in equation (3.1) to get f(x). G.f. is 1 - x * (1 - f(x)).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - Joerg Arndt, Jun 11 2013
Let c(n) = number of palindromic partitions of n whose greatest part has multiplicity 3; then c(n) = a(n-3) for n>=3. - Clark Kimberling, Mar 05 2014
From Gus Wiseman, Aug 22 2021: (Start)
Also the number of integer partitions of n - 1 whose parts cover an interval of positive integers starting with 2. These partitions are ranked by A339886. For example, the a(6) = 1 through a(16) = 5 partitions are:
  32  222  322  332   432   3322   3332   4332    4432    5432     43332
                2222  3222  22222  4322   33222   33322   33332    44322
                                   32222  222222  43222   43322    333222
                                                  322222  332222   432222
                                                          2222222  3222222
(End)

			1 + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 3*x^12 + 3*x^13 + ...
q + q^73 + q^121 + q^145 + q^169 + q^193 + 2*q^217 + 2*q^241 + 2*q^265 + ...
a(10)=2 because we have [7,3] and [5,5].
From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(22)=13 symmetric unimodal compositions of 22+3=25 where the maximal part appears three times:
01:  [ 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 2 3 3 3 2 1 1 1 1 1 1 ]
03:  [ 1 1 1 1 1 5 5 5 1 1 1 1 1 ]
04:  [ 1 1 1 1 2 2 3 3 3 2 2 1 1 1 1 ]
05:  [ 1 1 1 2 5 5 5 2 1 1 1 ]
06:  [ 1 1 2 2 2 3 3 3 2 2 2 1 1 ]
07:  [ 1 1 3 5 5 5 3 1 1 ]
08:  [ 1 1 7 7 7 1 1 ]
09:  [ 1 2 2 5 5 5 2 2 1 ]
10:  [ 1 4 5 5 5 4 1 ]
11:  [ 2 2 2 2 3 3 3 2 2 2 2 ]
12:  [ 2 3 5 5 5 3 2 ]
13:  [ 2 7 7 7 2 ]
(End)
From _Gus Wiseman_, Feb 16 2021: (Start)
The a(7) = 1 through a(19) = 8 partitions are the following (A..J = 10..19). The Heinz numbers of these partitions are given by A341449.
  7  53  9    55  B    75    D    77    F      97    H      99      J
         333  73  533  93    553  95    555    B5    755    B7      775
                       3333  733  B3    753    D3    773    D5      955
                                  5333  933    5533  953    F3      973
                                        33333  7333  B33    5553    B53
                                                     53333  7533    D33
                                                            9333    55333
                                                            333333  73333
(End)

J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. I

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from Alois P. Heinz)
C. Ballantine and M. Merca, Padovan numbers as sums over partitions into odd parts, Journal of Inequalities and Applications, (2016) 2016:1; doi.
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
Howard D. Grossman, Problem 228, Mathematics Magazine, 28 (1955), p. 160.
R. K. Guy, Two theorems on partitions, Math. Gaz., 42 (1958), 84-86. Math. Rev. 20 #3110.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.
James Mc Laughlin, Andrew V. Sills, and Peter Zimmer, Rogers-Ramanujan-Slater Type Identities, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

The ordered version is A000931.
Partitions with no ones are counted by A002865, ranked by A005408.
The even version is A035363, ranked by A066207.
The version for factorizations is A340101.
Partitions whose only even part is the smallest are counted by A341447.
The Heinz numbers of these partitions are given by A341449.
A000009 counts partitions into odd parts, ranked by A066208.
A025147 counts strict partitions with no 1's.
A025148 counts strict partitions with no 1's or 2's.
A026804 counts partitions whose smallest part is odd, ranked by A340932.
A027187 counts partitions with even length/maximum, ranks A028260/A244990.
A027193 counts partitions with odd length/maximum, ranks A026424/A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A058696 counts partitions of even numbers, ranked by A300061.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Cf. A000041, A003114, A039900, A160786, A257991/A257992, A264396, A300272, A339662, A339737, A339886.

a087897 = p [3,5..] where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

To get 128 terms: t4 := mul((1+x^(2^n)),n=0..7); t5 := mul((1+x^k),k=1..128): t6 := series(t5/t4,x,100); t7 := seriestolist(t6);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<3, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 11 2013

max = 65; f[x_] := Product[ 1/(1 - x^(2k+1)), {k, 1, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 16 2011, after Emeric Deutsch *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<3, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]] ]; a[n_] := b[n, n-1+Mod[n, 2]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 01 2015, after Alois P. Heinz *)
Flatten[{1, Table[PartitionsQ[n+1] - PartitionsQ[n], {n, 0, 80}]}] (* Vaclav Kotesovec, Dec 01 2015 *)
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&OddQ[Times@@#]&]],{n,0,30}] (* Gus Wiseman, Feb 16 2021 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - x) * eta(x^2 + A) / eta(x + A), n))} /* Michael Somos, Nov 13 2011 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A087897_T(n,k):
    if n==0: return 1
    if k<3 or n<0: return 0
    return A087897_T(n,k-2)+A087897_T(n-k,k)
def A087897(n): return A087897_T(n,n-(n&1^1)) # Chai Wah Wu, Sep 23 2023, after Alois P. Heinz

Expansion of q^(-1/24) * (1 - q) * eta(q^2) / eta(q) in powers of q.
Expansion of (1 - x) / chi(-x) in powers of x where chi() is a Ramanujan theta function.
G.f.: 1 + x^3 + x^5*(1 + x) + x^7*(1 + x)*(1 + x^2) + x^9*(1 + x)*(1 + x^2)*(1 + x^3) + ... [Glaisher 1876]. - Michael Somos, Jun 20 2012
G.f.: Product_{k >= 1} 1/(1-x^(2*k+1)).
G.f.: Product_{k >= 1, k not a power of 2} (1+x^k).
G.f.: Sum_{k >= 1} x^(3*k)/Product_{j = 1..k} (1 - x^(2*j)). - Emeric Deutsch, Mar 30 2006
a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (8 * 3^(3/4) * n^(5/4)) * (1 - (15*sqrt(3)/(8*Pi) + 11*Pi/(48*sqrt(3)))/sqrt(n) + (169*Pi^2/13824 + 385/384 + 315/(128*Pi^2))/n). - Vaclav Kotesovec, Aug 30 2015, extended Nov 04 2016
G.f.: 1/(1 - x^3) * Sum_{n >= 0} x^(5*n)/Product_{k = 1..n} (1 - x^(2*k)) = 1/((1 - x^3)*(1 - x^5)) * Sum_{n >= 0} x^(7*n)/Product_{k = 1..n} (1 - x^(2*k)) = ..., extending Deutsch's result dated Mar 30 2006. - Peter Bala, Jan 15 2021
G.f.: Sum_{n >= 0} x^(n*(2*n+1))/Product_{k = 2..2*n+1} (1 - x^k). (Set z = x^3 and q = x^2 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
a(2*n+1) = Sum{j>=1} A008284(n+1-j,2*j - 1) and a(2*n) = Sum{j>=1} A008284(n-j, 2*j). - Gregory L. Simay, Sep 22 2023

cp's OEIS Frontend

A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

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A360646 Square array A(n, k), n, k > 0, read by antidiagonals upwards; A(n, k) = A066208(n) * A066207(k).

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A374611 If n = A066207(i) * A066208(j) for some i, j > 0 then a(n) = A066207(j) * A066208(i).

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A027193 Number of partitions of n into an odd number of parts.

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A027187 Number of partitions of n into an even number of parts.

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A035363 Number of partitions of n into even parts.

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A066207 All primes that divide n are of the form prime(2k), where prime(k) is k-th prime.

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