This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A096373 #40 Jan 15 2025 10:59:31
%S A096373 0,1,0,2,1,3,3,6,5,11,11,17,20,30,33,49,56,77,92,122,143,190,225,287,
%T A096373 344,435,516,648,770,951,1134,1388,1646,2007,2376,2868,3395,4078,4807,
%U A096373 5749,6764,8042,9449,11187,13101,15463,18070,21236,24772,29021,33764
%N A096373 Number of partitions of n such that the least part occurs exactly twice.
%C A096373 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
%C A096373 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
%C A096373 Number of partitions of n+1 such that the two smallest parts differ by one. - _Giovanni Resta_, Mar 07 2014
%C A096373 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
%H A096373 Giovanni Resta, <a href="/A096373/b096373.txt">Table of n, a(n) for n = 1..1000</a>
%F A096373 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)).
%F A096373 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
%F A096373 a(n) = -p(n+3)+2*p(n+2)-p(n), p(n)=A000041(n). - _Mircea Merca_, Jul 10 2013
%F A096373 a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi / (12*sqrt(2)*n^(3/2)). - _Vaclav Kotesovec_, Jun 02 2018
%e A096373 a(6)=3 because we have [4,1,1], [3,3] and [2,2,1,1].
%e A096373 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 + ...
%e A096373 From _Gus Wiseman_, Oct 26 2018: (Start)
%e A096373 The a(2) = 1 through a(10) = 11 partitions where the least part occurs exactly twice (zero terms not shown):
%e A096373 (11) (22) (311) (33) (322) (44) (522) (55)
%e A096373 (211) (411) (511) (422) (711) (433)
%e A096373 (2211) (3211) (611) (4311) (622)
%e A096373 (3311) (5211) (811)
%e A096373 (4211) (32211) (3322)
%e A096373 (22211) (4411)
%e A096373 (5311)
%e A096373 (6211)
%e A096373 (33211)
%e A096373 (42211)
%e A096373 (222211)
%e A096373 The a(2) = 1 through a(10) = 11 partitions that cannot be grouped into pairs of distinct parts (zero terms not shown):
%e A096373 (11) (22) (2111) (33) (2221) (44) (3222) (55)
%e A096373 (1111) (3111) (4111) (2222) (6111) (3331)
%e A096373 (111111) (211111) (5111) (321111) (4222)
%e A096373 (221111) (411111) (7111)
%e A096373 (311111) (21111111) (222211)
%e A096373 (11111111) (331111)
%e A096373 (421111)
%e A096373 (511111)
%e A096373 (22111111)
%e A096373 (31111111)
%e A096373 (1111111111)
%e A096373 (End)
%p A096373 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
%t A096373 (* 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 *)
%t A096373 Table[Count[IntegerPartitions[n+2], p_ /; MemberQ[p, Length[p] + Min[p]]], {n, 50}] (* _Clark Kimberling_, Feb 27 2014 *)
%t A096373 p[n_, m_] := If[m == n, 1, If[m > n, 0, p[n, m] = Sum[p[n-m, k], {k, m, n}]]];
%t A096373 a[n_] := Sum[p[n+1-k, k+1], {k, n/2}]; Array[a, 100] (* _Giovanni Resta_, Mar 07 2014 *)
%o A096373 (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
%o A096373 (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 */
%Y A096373 Cf. A002865, A097091, A097092, A097093.
%Y A096373 Cf. A117989.
%Y A096373 Cf. A000070, A000569, A007717, A025065, A141268, A209816, A261049, A320328, A320891, A320921, A320922.
%K A096373 easy,nonn
%O A096373 1,4
%A A096373 _Vladeta Jovovic_, Jul 19 2004
%E A096373 Edited and extended by _Robert G. Wilson v_ and _Klaus Brockhaus_, Jul 21 2004