A240690 - OEIS

%I #22 Aug 08 2018 03:02:57

%S 0,1,1,2,3,4,7,8,14,16,26,30,47,54,81,95,136,161,224,266,361,431,571,

%T 684,891,1067,1369,1641,2077,2488,3116,3726,4623,5520,6790,8093,9884,

%U 11753,14262,16923,20415,24168,29006,34255,40920,48214,57344,67410,79863

%N Number of partitions p of n such that p contains fewer 1s than its conjugate.

%C a(n+1) = number of partitions p of n such that (# 1s in p) <= (#1s in conjugate(p)).

%H G. C. Greubel, <a href="/A240690/b240690.txt">Table of n, a(n) for n = 1..2500</a>

%F 2*a(n) + A240691(n) = A000041(n) for n >= 1.

%F a(n) + a(n+1) = A000041(n). - _Omar E. Pol_, Mar 07 2015

%F G.f.: (-1 + Product_{k>0} (1 - x^k)^(-1)) * x / (1 + x). - _Michael Somos_, Mar 16 2015

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)). - _Vaclav Kotesovec_, Jun 02 2018

%e a(6) counts these 4 partitions: 6, 51, 42, 411, of which the respective conjugates are 111111, 21111, 2211, 3111.

%e G.f. = x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 8*x^8 + 14*x^9 + 16*x^10 + ...

%t z = 53; f[n_] := f[n] = IntegerPartitions[n]; c[p_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[p];  (* conjugate of partition p *)

%t Table[Count[f[n], p_ /; Count[p, 1] < Count[c[p], 1]], {n, 1, z}]  (* A240690 *)

%t Table[Count[f[n], p_ /; Count[p, 1] <= Count[c[p], 1]], {n, 1, z}]  (* A240690(n+1) *)

%t Table[Count[f[n], p_ /; Count[p, 1] == Count[c[p], 1]], {n, 1, z}] (* A240691 *)

%t a[ n_] := SeriesCoefficient[ (-1 + 1 / QPochhammer[ x]) x / (1 + x), {x, 0, n}]; (* _Michael Somos_, Mar 16 2015 *)

%o (PARI) q='q+O('q^60); concat([0], Vec((-1 + 1/eta(q))*q/(1+q))) \\ _G. C. Greubel_, Aug 07 2018

%Y Cf. A240691, A000041.

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Apr 11 2014