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%I #22 Jan 25 2022 10:25:57
%S 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,
%T 272,101,373,135,508,176,684,231,915,297,1212,385,1597,490,2087,627,
%U 2714,792,3506,1002,4508,1255,5763,1575,7338,1958,9296,2436,11732,3010,14742
%N Number of partitions of n with at most one odd part.
%C From _Gus Wiseman_, Jan 21 2022: (Start)
%C 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:
%C 1 11 21 22 32 33 43 44 54
%C 111 1111 221 2211 331 2222 441
%C 2111 111111 2221 3311 3222
%C 11111 3211 221111 3321
%C 22111 11111111 4311
%C 211111 22221
%C 1111111 33111
%C 222111
%C 321111
%C 2211111
%C 21111111
%C 111111111
%C (End)
%H Alois P. Heinz, <a href="/A100824/b100824.txt">Table of n, a(n) for n = 0..1000</a>
%F 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).
%F 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
%F a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - _David A. Corneth_, Jan 23 2022
%e From _Gus Wiseman_, Jan 21 2022: (Start)
%e The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (21) (22) (32) (42) (43) (44) (54)
%e (41) (222) (52) (62) (63)
%e (221) (61) (422) (72)
%e (322) (2222) (81)
%e (421) (432)
%e (2221) (441)
%e (522)
%e (621)
%e (3222)
%e (4221)
%e (22221)
%e (End)
%p 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)
%t 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 *)
%t Table[Length[Select[IntegerPartitions[n],Count[#,_?OddQ]<=1&]],{n,0,30}] (* _Gus Wiseman_, Jan 21 2022 *)
%o (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
%Y Cf. A008951, A000070, A000097, A000098, A000710.
%Y The case of alternating sum 0 (equality) is A000070.
%Y A multiplicative version is A339846.
%Y These partitions are ranked by A349150, conjugate A349151.
%Y A000041 = integer partitions, strict A000009.
%Y A027187 = partitions of even length, strict A067661, ranked by A028260.
%Y A027193 = partitions of odd length, ranked by A026424.
%Y A058695 = partitions of odd numbers.
%Y A103919 = partitions by sum and alternating sum (reverse: A344612).
%Y A277103 = partitions with the same number of odd parts as their conjugate.
%Y Cf. A000984, A001791, A008549, A097805, A119620, A182616, A236559, A236913, A236914, A304620, A344607, A345958, A347443.
%K easy,nonn
%O 0,4
%A _Vladeta Jovovic_, Jan 13 2005
%E More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005
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