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A238395 Number of partitions of n that sorted in increasing order contain a part k in position k for some k. 3
0, 1, 1, 2, 4, 5, 8, 12, 18, 25, 34, 47, 65, 88, 118, 154, 203, 263, 343, 442, 568, 721, 914, 1149, 1445, 1807, 2255, 2800, 3468, 4270, 5250, 6425, 7855, 9566, 11635, 14103, 17068, 20584, 24784, 29754, 35670, 42653, 50934, 60688, 72212, 85742, 101662, 120293 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Note that considering partitions in standard decreasing order, we obtain A001522.

LINKS

Giovanni Resta and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Giovanni Resta)

FORMULA

a(n) + A238394(n) = p(n) = A000041(n).

EXAMPLE

a(6) = 11 - 3 = 8, because of the 11 partitions of 6 only 3 do not contain a 1 in position 1, a 2 in position 2, or a 3 in position 3, namely (3,3), (2,4) and (6).

MAPLE

b:= proc(n, i) option remember; `if`(n=0, [0, 1],

      `if`(i<1, [0$2], b(n, i-1) +`if`(i>n, 0,

      (p->[p[1] +coeff(p[2], x, i-1), expand(x*(p[2]-

       coeff(p[2], x, i-1)*x^(i-1)))])(b(n-i, i)))))

    end:

a:= n-> b(n$2)[1]:

seq(a(n), n=0..70);  # Alois P. Heinz, Feb 26 2014

MATHEMATICA

a[n_] := Length@ Select[IntegerPartitions@ n, MemberQ[ Reverse@# - Range@ Length@#, 0] &]; Array[a, 30]

(* Second program: *)

b[n_, i_] := b[n, i] = If[n==0, {0, 1}, If[i<1, {0, 0}, b[n, i-1] + If[i>n, 0, Function[p, {p[[1]] + Coefficient[p[[2]], x, i-1], x*(p[[2]] - Coefficient[p[[2]], x, i-1]*x^(i-1))}][b[n-i, i]]]]]; a[n_] := b[n, n][[1]]; Table[a[n], {n, 0, 70}] (* Jean-Fran├žois Alcover, Aug 29 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A238394, A064428, A001522.

Sequence in context: A102186 A039842 A188216 * A116901 A244487 A290962

Adjacent sequences:  A238392 A238393 A238394 * A238396 A238397 A238398

KEYWORD

nonn

AUTHOR

Giovanni Resta, Feb 26 2014

STATUS

approved

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Last modified July 25 14:49 EDT 2021. Contains 346290 sequences. (Running on oeis4.)