A116860 -id:A116860 - cp's OEIS frontend

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

Clark Kimberling

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

			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 + ...

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

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

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

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 *)

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

A092319 Sum of smallest parts of all partitions of n into odd distinct parts.

Original entry on oeis.org

1, 0, 3, 1, 5, 1, 7, 4, 10, 4, 12, 9, 15, 9, 20, 17, 23, 17, 28, 27, 36, 28, 41, 43, 50, 44, 62, 62, 71, 66, 84, 91, 103, 96, 119, 127, 139, 137, 167, 178, 191, 192, 223, 241, 266, 264, 302, 331, 351, 360, 411, 439, 469, 485, 542, 587, 628, 646, 714, 773, 819, 854, 945

Offset: 1

Vladeta Jovovic, Feb 15 2004

a(n) = Sum_{k>=0} A116860(n,k). - Emeric Deutsch, Feb 27 2006

			a(13)=15 because the partitions of 13 into distinct odd parts are [13],[9,3,1] and [7,5,1], with sum of the smallest terms 13+1+1=15.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A092316.

Cf. A116860.

f:=sum((2*n-1)*x^(2*n-1)*product(1+x^(2*k+1),k=n..40),n=1..40): fser:=simplify(series(f,x=0,66)): seq(coeff(fser,x^n),n=1..63); # Emeric Deutsch, Feb 27 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+2)+b(n-i, i+2)))
    end:
a:= n-> add(`if`(j::odd, j*b(n-j, j+2), 0), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

nmax = 60; Rest[CoefficientList[Series[Sum[(2*k - 1)*x^(2*k - 1) * Product[1 + x^(2*j + 1), {j, k, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 28 2016 *)

G.f.: Sum((2*n-1)*x^(2*n-1)*Product(1+x^(2*k+1), k = n .. infinity), n = 1 .. infinity).

a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/6)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, May 20 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004

cp's OEIS Frontend

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

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