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A100824
Number of partitions of n with at most one odd part.
10
1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742
OFFSET
0,4
COMMENTS
From Gus Wiseman, Jan 21 2022: (Start)
Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:
1 11 21 22 32 33 43 44 54
111 1111 221 2211 331 2222 441
2111 111111 2221 3311 3222
11111 3211 221111 3321
22111 11111111 4311
211111 22221
1111111 33111
222111
321111
2211111
21111111
111111111
(End)
LINKS
FORMULA
G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).
a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - Vaclav Kotesovec, Mar 07 2016
a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - David A. Corneth, Jan 23 2022
EXAMPLE
From Gus Wiseman, Jan 21 2022: (Start)
The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (22) (32) (42) (43) (44) (54)
(41) (222) (52) (62) (63)
(221) (61) (422) (72)
(322) (2222) (81)
(421) (432)
(2221) (441)
(522)
(621)
(3222)
(4221)
(22221)
(End)
MAPLE
seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i), i=1..100), x, 100), polynom), x, n), n=0..60); (C. Ronaldo)
MATHEMATICA
nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n], Count[#, _?OddQ]<=1&]], {n, 0, 30}] (* Gus Wiseman, Jan 21 2022 *)
PROG
(PARI) a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022
CROSSREFS
The case of alternating sum 0 (equality) is A000070.
A multiplicative version is A339846.
These partitions are ranked by A349150, conjugate A349151.
A000041 = integer partitions, strict A000009.
A027187 = partitions of even length, strict A067661, ranked by A028260.
A027193 = partitions of odd length, ranked by A026424.
A058695 = partitions of odd numbers.
A103919 = partitions by sum and alternating sum (reverse: A344612).
A277103 = partitions with the same number of odd parts as their conjugate.
Sequence in context: A007439 A238622 A096441 * A163227 A325690 A238779
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Jan 13 2005
EXTENSIONS
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005
STATUS
approved