A096765 Number of partitions of n into distinct parts, the least being 1.

0, 1, 0, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 17, 21, 25, 29, 35, 41, 48, 56, 66, 76, 89, 103, 119, 137, 159, 181, 209, 239, 273, 312, 356, 404, 460, 522, 591, 669, 757, 853, 963, 1085, 1219, 1371, 1539, 1725, 1933, 2164, 2418, 2702, 3016, 3362, 3746, 4171, 4637

Offset

0,7

Comments

The old entry with this sequence number was a duplicate of A071569.

a(n) is also the total number of 1's in all partitions of n into distinct parts. For n=6 there are partitions [6], [5,1], [4,2], [3,2,1] and only two contain a 1, hence a(6) = 2. - T. Amdeberhan, May 13 2012

a(n), n > 1 is the Euler transform of [0,1,1] joined with period [0,1]. - Georg Fischer, Aug 15 2020

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = A025147(n-1), n>1. - R. J. Mathar, Jul 31 2008

G.f.: x*Product_{j=2..infinity} (1+x^j). - R. J. Mathar, Jul 31 2008

G.f.: x / ((1 + x) * Product_{k>0} (1 - x^(2*k-1))). - Michael Somos, Sep 10 2016

G.f.: Sum_{k>0} x^(k*(k+1)/2) / Product_{j=1..k-1} (1 - x^j). - Michael Somos, Sep 10 2016

Example

G.f. = x + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 5*x^10 + 5*x^11 + ...

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-1)*(i+2)/2<n, 0,
       add(b(n-i*j, i-1), j=0..min(1, n/i))))
    end:
a:= n-> `if`(n<1, 0, b(n-1$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 07 2014
# Using the function EULER from Transforms (see link at the bottom of the page).
[0, 1, op(EULER([0, 1, seq(irem(n, 2), n=1..56)]))]; # Peter Luschny, Aug 19 2020

Mathematica

p[_, 0] = 1; p[k_, n_] := p[k, n] = If[n < k, 0, p[k+1, n-k] + p[k+1, n]]; a[n_] := p[2, n-1]; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Apr 17 2014, after Reinhard Zumkeller *)
a[ n_] := SeriesCoefficient[ x / ((1 + x) Product[ 1 - x^j, {j, 1, n, 2}]), {x, 0, n}]; (* Michael Somos, Sep 10 2016 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[  Sum[ x^(k (k + 1)/2) / Product[ 1 - x^j, {j, 1, k - 1}], {k, 1, Quotient[-1 + Sqrt[8 n + 1], 2]}], {x, 0, n}]]; (* Michael Somos, Sep 10 2016 *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 1], {n, 66}]] (* Robert Price, Jun 13 2020 *)

Prog

(PARI) {a(n) = if( n<1, 0, polcoeff( x / ((1 + x) * prod(k=1, (n+1)\2, 1 - x^(2*k-1), 1 + O(x^n))), n))}; /* Michael Somos, Sep 10 2016 */

Crossrefs

Cf. A025147, A071569.

Cf. A096749 (least=2), A022824 (3), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

Sequence in context: A029018 A238217 A185226 * A025147 A364613 A032230

Adjacent sequences: A096762 A096763 A096764 * A096766 A096767 A096768

Keyword

nonn

Author

N. J. A. Sloane, Sep 28 2008

Status

approved