A027349 Number of partitions of n into distinct odd parts, the least being 1.

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 4, 4, 5, 6, 6, 6, 8, 8, 9, 9, 11, 12, 13, 13, 16, 17, 18, 19, 22, 24, 25, 27, 30, 33, 35, 37, 41, 46, 47, 51, 56, 61, 64, 69, 75, 82, 86, 92, 100, 109, 114, 122, 133, 143, 151, 161, 174, 187, 198

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Column 1 of A116860. - Emeric Deutsch, Feb 27 2006

Also number of partitions of n such that the largest part occurs exactly once and each number smaller than the largest part occurs an even nonzero number of times. Example: a(17)=3 because we have [3,2,2,2,2,2,2,1,1],[3,2,2,2,2,1,1,1,1,1,1] and [3,2,2,1,1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 13 2006

a(n) is the number of symmetric stack polyominoes of area n with square core. The core of a stack is the set of all maximal columns. The core is a square when the number of columns is equal to their height. Equivalently, a(n) is the number of symmetric unimodal compositions of n, where the number of the parts of maximum value equal the maximum value itself. For instance, for n = 20, we have the following stacks: (2,4,4,4,4,2), (1,1,4,4,4,4,1,1), (1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1). - Emanuele Munarini, Apr 08 2011

Examples

			a(17) = 3 because we have [13,3,1], [11,5,1] and [9,7,1].
G.f. = x + x^4 + x^6 + x^8 + x^9 + x^10 + x^11 + x^12 + 2*x^13 + x^14 + 2*x^15 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Abdulaziz Alanazi, Augustine O. Munagi, and Andrew V. Sills, "Overpartitionized" Rogers-Ramanujan type identities, arXiv:2501.17826 [math.CO], 2025. See pp. 5, 13.
• Jason Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 3, with k=1. - _N. J. A. Sloane_, Aug 31 2014
• Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Crossrefs

Cf. A000700, A116860, A001523, A096441, A188674.

Programs

Maple

N := 100; t1 := series(mul(1+x^(2*k+1),k=1..N),x,N); A027349 := proc(n) coeff(t1,x,n); end;

Mathematica

a[n_]:=CoefficientList[Series[1+Sum[x^((k+1)^2)/Product[(1-x^(2i)),{i,1,k}],{k,0,n}],{x,0,n}],x] (* Emanuele Munarini, Apr 08 2011 *)
a[ n_] := SeriesCoefficient[ x QHypergeometricPFQ[ {}, {}, x^2, -x^3], {x, 0, n}]; (* Michael Somos, Feb 02 2015 *)
nmax = 100; Rest[CoefficientList[Series[x/(1+x) * Product[1+x^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)
(2/((1 + x) QPochhammer[-1, -x]) + O[x]^70)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)

Formula

G.f.: x*Product_{i>=2} 1+x^(2*i-1). - Emeric Deutsch, Feb 27 2006

G.f.: (Sum_{k>=1} x^(k^2))/Product_{j=1..k-1} 1-x^(2*j). - Emeric Deutsch, Mar 13 2006

a(n) ~ exp(Pi*sqrt(n/6)) / (2^(11/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

G.f.: 2/((1 + x)*(-1; -x)inf), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 22 2016

If n > 1, a(n) = A000700(n - 1) - a(n - 1). - Álvar Ibeas, Aug 03 2020

G.f.: x*Sum_{n >= 0} x^(n*(n+2))/Product_{k = 1..n} (1 - x^(2*k)) = x*(1 + x^3) * Sum_{n >= 0} x^(n*(n+4))/Product_{k = 1..n} (1 - x^(2*k)) = x*(1 + x^3)*(1 + x^5) * Sum_{n >= 0} x^(n*(n+6))/ Product_{k = 1..n} (1 - x^(2*k)) = .... - Peter Bala, Jan 15 2021