A096373 Number of partitions of n such that the least part occurs exactly twice.

0, 1, 0, 2, 1, 3, 3, 6, 5, 11, 11, 17, 20, 30, 33, 49, 56, 77, 92, 122, 143, 190, 225, 287, 344, 435, 516, 648, 770, 951, 1134, 1388, 1646, 2007, 2376, 2868, 3395, 4078, 4807, 5749, 6764, 8042, 9449, 11187, 13101, 15463, 18070, 21236, 24772, 29021, 33764

Offset: 1

Author

Vladeta Jovovic, Jul 19 2004

Keywords

• easy
• nonn

Comments

Also number of partitions of n such that the difference between the two largest distinct parts is 2 (it is assumed that 0 is a part in each partition). Example: a(6)=3 because we have [4,2], [3,1,1,1] and [2,2,2]. - Emeric Deutsch, Apr 08 2006

Number of partitions p of n+2 such that min(p) + (number of parts of p) is a part of p. - Clark Kimberling, Feb 27 2014

Number of partitions of n+1 such that the two smallest parts differ by one. - Giovanni Resta, Mar 07 2014

Also the number of integer partitions of n with an even number of parts that cannot be grouped into pairs of distinct parts. These are also integer partitions of n with an even number of parts whose greatest multiplicity is greater than half the number of parts. - Gus Wiseman, Oct 26 2018

Examples

			a(6)=3 because we have [4,1,1], [3,3] and [2,2,1,1].
G.f. = x^2 + 2*x^4 + x^5 + 3*x^6 + 3*x^7 + 6*x^8 + 5*x^9 + 11*x^10 + 11*x^11 + ...
From _Gus Wiseman_, Oct 26 2018: (Start)
The a(2) = 1 through a(10) = 11 partitions where the least part occurs exactly twice (zero terms not shown):
  (11)  (22)   (311)  (33)    (322)   (44)     (522)    (55)
        (211)         (411)   (511)   (422)    (711)    (433)
                      (2211)  (3211)  (611)    (4311)   (622)
                                      (3311)   (5211)   (811)
                                      (4211)   (32211)  (3322)
                                      (22211)           (4411)
                                                        (5311)
                                                        (6211)
                                                        (33211)
                                                        (42211)
                                                        (222211)
The a(2) = 1 through a(10) = 11 partitions that cannot be grouped into pairs of distinct parts (zero terms not shown):
  (11) (22)   (2111) (33)     (2221)   (44)       (3222)     (55)
       (1111)        (3111)   (4111)   (2222)     (6111)     (3331)
                     (111111) (211111) (5111)     (321111)   (4222)
                                       (221111)   (411111)   (7111)
                                       (311111)   (21111111) (222211)
                                       (11111111)            (331111)
                                                             (421111)
                                                             (511111)
                                                             (22111111)
                                                             (31111111)
                                                             (1111111111)
(End)
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002865, A097091, A097092, A097093.

Cf. A117989.

Cf. A000070, A000569, A007717, A025065, A141268, A209816, A261049, A320328, A320891, A320921, A320922.

Programs

Maple

g:=sum(x^(2*k)/product(1-x^j,j=k+1..80),k=1..70): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # Emeric Deutsch, Apr 08 2006

Mathematica

(* do first *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Block[{p = Partitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 3]; If[ q[[1]] != q[[3]] && q[[2]] == q[[3]], c++ ]; k++ ]; c]; Table[ f[n], {n, 51}] (* Robert G. Wilson v, Jul 23 2004 *)
Table[Count[IntegerPartitions[n+2], p_ /; MemberQ[p, Length[p] + Min[p]]], {n, 50}] (* Clark Kimberling, Feb 27 2014 *)
p[n_, m_] := If[m == n, 1, If[m > n, 0, p[n, m] = Sum[p[n-m, k], {k, m, n}]]];
a[n_] := Sum[p[n+1-k, k+1], {k, n/2}]; Array[a, 100] (* Giovanni Resta, Mar 07 2014 *)

PARI

{q=sum(m=1,100,x^(2*m)/prod(i=m+1,100,1-x^i,1+O(x^60)),1+O(x^60));for(n=1,51,print1(polcoeff(q,n),","))} \\ Klaus Brockhaus, Jul 21 2004

PARI

{a(n) = if( n<0, 0, polcoeff( ( 1 - (1 - x - x^2) / eta(x + x^4 * O(x^n)) ) * (1 - x) / x^3, n))} /* Michael Somos, Feb 28 2014 */

Formula

G.f.: Sum_{m>0} (x^(2*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m)/Product_{i>m} (1-x^i)).

G.f.: Sum_{k>=1} (x^(2*k-2)*(1-x^(k-1))/Product_{j=1..k} (1-x^j)). - Emeric Deutsch, Apr 08 2006

a(n) = -p(n+3)+2*p(n+2)-p(n), p(n)=A000041(n). - Mircea Merca, Jul 10 2013

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (12*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Jun 02 2018

Extensions

Edited and extended by Robert G. Wilson v and Klaus Brockhaus, Jul 21 2004