A212545 Number of partitions of n containing at least one part m-5 if m is the largest part.

0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 39, 52, 75, 98, 137, 175, 236, 300, 393, 493, 635, 787, 997, 1227, 1531, 1869, 2309, 2796, 3420, 4119, 4994, 5979, 7201, 8574, 10260, 12164, 14470, 17082, 20225, 23778, 28025, 32838, 38542, 45011, 52642, 61286, 71434, 82937

Offset

5,5

Links

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Formula

G.f.: Sum_{i>0} x^(2*i+5) / Product_{j=1..5+i} (1-x^j).

Example

a(7) = 1: [6,1].
a(8) = 1: [6,1,1].
a(9) = 3: [6,1,1,1], [6,2,1], [7,2].
a(10) = 4: [6,1,1,1,1], [6,2,1,1], [6,3,1], [7,2,1].
a(11) = 8: [6,1,1,1,1,1], [6,2,1,1,1], [6,2,2,1], [6,3,1,1], [6,4,1], [7,2,1,1], [7,2,2], [8,3].

Maple

b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-5, min(n-2*m-5, m+5)), m=1..(n-5)/2):
seq(a(n), n=5..60);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 5, Min[n - 2 m - 5, m + 5]], {m, 1, (n - 5)/2}];
a /@ Range[5, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Crossrefs

Column k=5 of A212551.

Sequence in context: A288566 A084421 A271723 * A357878 A358910 A212546

Adjacent sequences: A212542 A212543 A212544 * A212546 A212547 A212548

Keyword

nonn

Author

Alois P. Heinz, May 20 2012

Status

approved