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A240690 Number of partitions p of n such that p contains fewer 1s than its conjugate. 4
0, 1, 1, 2, 3, 4, 7, 8, 14, 16, 26, 30, 47, 54, 81, 95, 136, 161, 224, 266, 361, 431, 571, 684, 891, 1067, 1369, 1641, 2077, 2488, 3116, 3726, 4623, 5520, 6790, 8093, 9884, 11753, 14262, 16923, 20415, 24168, 29006, 34255, 40920, 48214, 57344, 67410, 79863 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

Table of n, a(n) for n=1..49.

FORMULA

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

a(n) + a(n+1) = A000041(n). - Omar E. Pol, Mar 07 2015

G.f.: (-1 + Product_{k>0} (1 - x^k)^(-1)) * x / (1 + x). - Michael Somos, Mar 16 2015

EXAMPLE

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

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 + ...

MATHEMATICA

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

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

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

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

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

CROSSREFS

Cf. A240691, A000041.

Sequence in context: A215914 A006049 A084541 * A113050 A278180 A015927

Adjacent sequences:  A240687 A240688 A240689 * A240691 A240692 A240693

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 11 2014

STATUS

approved

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Last modified August 21 07:08 EDT 2017. Contains 290862 sequences.