A022828 -id:A022828 - cp's OEIS frontend

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

N. J. A. Sloane, Sep 28 2008

nonn

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

			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 + ...

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

Cf. A025147, A071569.

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

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-1)*(i+2)/2 `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

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 *)

{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 */

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

A096749 Number of partitions of n into distinct parts, the least being 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 15, 17, 20, 24, 28, 32, 38, 44, 51, 59, 68, 78, 91, 103, 118, 136, 155, 176, 201, 228, 259, 294, 332, 375, 425, 478, 538, 607, 681, 764, 858, 961, 1075, 1203, 1343, 1499, 1673, 1863, 2073, 2308, 2564, 2847, 3161, 3504

Offset: 0

N. J. A. Sloane, Sep 28 2008

nonn

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

a(n), n>2 is the Euler transform of [0,0,1,1,1] joined with period [0,1]. - Georg Fischer, Aug 15 2020

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A025147, A071569.

Cf. A096765 (least=1), A022824 (3), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-2)*(i+3)/2 `if`(n<2, 0, b(n-2$2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 07 2014
# Using the function EULER from Transforms (see link at the bottom of the page).
[0,0,1,op(EULER([0,0,1,1,seq(irem(n,2),n=1..57)]))]; # Peter Luschny, Aug 19 2020

b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i-2)*(i+3)/2Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 2], {n, 66}]] (* Robert Price, Jun 13 2020 *)

G.f.: x^2*Product_{j>=3} (1+x^j). - R. J. Mathar, Jul 31 2008
a(n) = A025148(n-2), n>1. - R. J. Mathar, Sep 30 2008
G.f.: Sum_{k>=1} x^(k*(k + 3)/2) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 24 2020

A026824 Number of partitions of n into distinct parts, the least being 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 17, 20, 23, 27, 31, 36, 41, 47, 55, 62, 71, 81, 93, 105, 120, 135, 154, 174, 197, 221, 251, 281, 317, 356, 400, 447, 502, 561, 628, 701, 782, 871, 972, 1081, 1202, 1336, 1483, 1645, 1825, 2021, 2237, 2476

Offset: 0

Clark Kimberling

nonn

Also, number of partitions of n such that if k is the largest part, then k occurs exactly 3 times and each of the numbers 1,2,...,k-1 occur at least once (these are the conjugates of the partitions described in the definition). Example: a(14)=3 because we have [3,3,3,2,2,1],[3,3,3,2,1,1,1] and [2,2,2,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 17 2006

For n > 3, a(n) is the Euler transform of [0,0,0,1,1,1,1] joined with the period 2 sequence [0,1, ...]. - Georg Fischer, Aug 18 2020

			a(14) = 3 because we have [11,3], [7,4,3] and [6,5,3].

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

Cf. A025147, A025149.

Cf. A096765 (least=1), A096749 (2), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

g:=x^3*product(1+x^j,j=4..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..59); # Emeric Deutsch, Apr 17 2006
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-3)*(i+4)/2 `if`(n<3, 0, b(n-3$2)):
seq(a(n), n=0..60); # Alois P. Heinz, Feb 07 2014

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[(i-3)(i+4)/2 < n, 0, Sum[b[n-i*j, i-1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n<3, 0, b[n-3, n-3]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[x^3/((1+x)*(1+x^2)*(1+x^3)) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 3], {n, 1, 66}]] (* Robert Price, Jun 13 2020 *)

From Emeric Deutsch, Apr 17 2006: (Start)
G.f.: (x^3)*Product_{j=4..infinity} (1+x^j).
G.f.: Sum_{k=1..infinity} x^(k*(k+5)/2)/(Product_{j=1..k-1} (1-x^j)). (End)
a(n) = A025149(n-3), n>3. - R. J. Mathar, Jul 31 2008
a(n) ~ exp(Pi*sqrt(n/3)) / (32*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

More terms from Emeric Deutsch, Apr 17 2006

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

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A096749 Number of partitions of n into distinct parts, the least being 2.

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A026824 Number of partitions of n into distinct parts, the least being 3.

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