A032021 -id:A032021 - cp's OEIS frontend

A219107 Number of compositions (ordered partitions) of n into distinct prime parts.

1, 0, 1, 1, 0, 3, 0, 3, 2, 2, 8, 1, 8, 3, 8, 8, 10, 25, 16, 9, 16, 38, 16, 61, 18, 62, 46, 66, 160, 91, 138, 99, 70, 122, 306, 126, 314, 151, 362, 278, 588, 901, 602, 303, 654, 1142, 888, 1759, 892, 1226, 950, 2160, 1230, 3379, 1444, 2372, 2100, 4644, 7416

Offset: 0

Alois P. Heinz, Nov 11 2012

nonn

a(0) = 0 iff n in {1,4,6}.

			a(5) = 3: [2,3], [3,2], [5].
a(7) = 3: [2,5], [5,2], [7].
a(8) = 2: [3,5], [5,3].
a(9) = 2: [2,7], [7,2].
a(10) = 8: [2,3,5], [2,5,3], [3,2,5], [3,5,2], [5,2,3], [5,3,2], [3,7], [7,3].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000142, A000586, A032021, A218396.

with(numtheory):
b:= proc(n, i) b(n, i):=
      `if`(n=0, [1], `if`(i<1, [], zip((x, y)->x+y, b(n, i-1),
       [0, `if`(ithprime(i)>n, [], b(n-ithprime(i), i-1))[]], 0)))
    end:
a:= proc(n) local l; l:= b(n, pi(n));
      a(n):= add(l[i]*(i-1)!, i=1..nops(l))
    end:
seq(a(n), n=0..70);
# second Maple program:
s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i, t) option remember; `if`(s(i)`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
    end:
a:= n-> b(n, numtheory[pi](n), 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 30 2020

zip = With[{m=Max[Length[#1], Length[#2]]}, PadRight[#1, m]+PadRight[#2, m] ]&;
b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, b[n, i-1] ~zip~ Join[{0}, If[Prime[i] > n, {}, b[n - Prime[i], i-1]]], {0}]];
a[n_] := Module[{l}, l = b[n, PrimePi[n]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 24 2017, adapted from Maple *)

a(n) = Sum_{k=0..A024936(n)} A219180(n,k)*k!.

A331844 Number of compositions (ordered partitions) of n into distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 3, 8, 0, 0, 8, 30, 0, 0, 0, 2, 6, 1, 2, 6, 24, 2, 8, 6, 0, 0, 8, 30, 0, 0, 7, 32, 24, 2, 8, 30, 120, 6, 24, 2, 6, 0, 8, 36, 24, 1, 34, 150, 0, 2, 12, 30, 24, 0, 2, 38, 150, 0, 12, 78, 144, 2

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(14) = 6 because we have [9,4,1], [9,1,4], [4,9,1], [4,1,9], [1,9,4] and [1,4,9].

Cf. A000290, A006456, A032020, A032021, A032022, A033461, A218396, A219107, A331843, A331845, A331846, A331847.

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..82);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n, Sqrt[n] // Floor, 0];
a /@ Range[0, 82] (* Jean-François Alcover, Oct 29 2020, after Alois P. Heinz *)

A331843 Number of compositions (ordered partitions) of n into distinct triangular numbers.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 0, 2, 7, 2, 0, 2, 6, 1, 4, 6, 2, 12, 24, 3, 8, 0, 8, 32, 6, 2, 13, 26, 6, 34, 36, 0, 32, 150, 3, 20, 50, 14, 54, 126, 32, 32, 12, 55, 160, 78, 122, 44, 174, 4, 72, 294, 36, 201, 896, 128, 62, 180, 176, 164, 198, 852, 110, 320, 159, 212, 414

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(10) = 7 because we have [10], [6, 3, 1], [6, 1, 3], [3, 6, 1], [3, 1, 6], [1, 6, 3] and [1, 3, 6].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Index entries for sequences related to compositions

Cf. A000217, A023361, A024940, A032020, A032021, A032022, A218396, A219107, A331844, A331845, A331846, A331847.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), 1+h(n-1), h(n-1)))
    end:
b:= proc(n, i, p) option remember; (t->
      `if`(t*(i+2)/3n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
    end:
a:= n-> b(n, h(n), 0):
seq(a(n), n=0..73);  # Alois P. Heinz, Jan 31 2020

h[n_] := h[n] = If[n<1, 0, If[IntegerQ @ Sqrt[8n+1], 1 + h[n-1], h[n-1]]];
b[n_, i_, p_] := b[n, i, p] = Function[t, If[t (i + 2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t>n, 0, b[n - t, i - 1, p + 1]]]]][(i(i + 1)/2)];
a[n_] := b[n, h[n], 0];
a /@ Range[0, 73] (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

A218396 Number of compositions of n into distinct (nonzero) Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 8, 2, 9, 8, 8, 32, 6, 9, 32, 8, 38, 30, 32, 150, 6, 33, 32, 32, 158, 30, 38, 174, 30, 176, 150, 150, 870, 24, 33, 152, 32, 182, 150, 158, 894, 30, 182, 174, 174, 1014, 144, 176, 990, 150, 1014, 864, 870, 5904, 24, 153, 152, 152, 902, 150, 182, 1014, 150, 1022, 894, 894, 6054, 144

Offset: 0

Joerg Arndt, Oct 28 2012

nonn

			There are a(37)=182 such compositions of 37. Each of the 6 partitions of 37 into distinct Fibonacci numbers corresponds to m! compositions (where m is the number of parts):
  #:  partition      ( m! compositions)
  1:  1 2 5 8 21     (120 compositions)
  2:  1 2 13 21      ( 24 compositions)
  3:  1 2 34         (  6 compositions)
  4:  3 5 8 21       ( 24 compositions)
  5:  3 13 21        (  6 compositions)
  6:  3 34           (  2 compositions)
The number of compositions is 120 + 24 + 6 + 24 + 6 + 2 = 182.

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms 0..200 from Joerg Arndt)

Cf. A032021 (compositions into distinct odd numbers).
Cf. A000119 (partitions into distinct nonzero Fibonacci numbers), A000700 (partitions into distinct odd numbers).
Cf. A076739 (compositions into Fibonacci numbers).

A331845 Number of compositions (ordered partitions) of n into distinct cubes.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 6, 24

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

			a(36) = 6 because we have [27,8,1], [27,1,8], [8,27,1], [8,1,27], [1,27,8] and [1,8,27].

Cf. A000578, A023358, A032020, A032021, A032022, A218396, A219107, A279329, A331843, A331844, A331846, A331847.

b:= proc(n, i, p) option remember;
      `if`((i*(i+1)/2)^2n, 0, b(n-i^3, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, iroot(n, 3), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[(i(i+1)/2)^2 < n, 0, If[n == 0, p!, If[i^3 > n, 0, b[n-i^3, i-1, p+1]] + b[n, i-1, p]]];
a[n_] := b[n, Floor[n^(1/3)], 0];
a /@ Range[0, 100] (* Jean-François Alcover, Oct 31 2020, after Alois P. Heinz *)

A337547 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 3.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 1, 2, 0, 1, 4, 6, 1, 4, 6, 1, 6, 12, 1, 6, 18, 25, 8, 24, 25, 8, 30, 49, 10, 42, 73, 10, 48, 121, 132, 60, 145, 132, 72, 217, 254, 84, 265, 374, 96, 361, 616, 114, 433, 856, 846, 553, 1218, 864, 649, 1578, 1602, 817, 2180, 2340, 937, 2780, 3798, 1129, 3622

Offset: 0

Ilya Gutkovskiy, Nov 22 2020

nonn

			a(12) = 6 because we have [7, 4, 1], [7, 1, 4], [4, 7, 1], [4, 1, 7], [1, 7, 4] and [1, 4, 7].

Index entries for sequences related to compositions

Cf. A000930, A016777, A032020, A032021, A035382, A261612, A337548, A339059, A339060.

nmax = 65; CoefficientList[Series[Sum[k! x^(k (3 k - 1)/2)/Product[1 - x^(3 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(3*k - 1)/2) / Product_{j=1..k} (1 - x^(3*j)).

A337548 Number of compositions (ordered partitions) of n into distinct parts congruent to 2 mod 3.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 4, 1, 6, 4, 1, 6, 6, 1, 12, 6, 1, 18, 8, 25, 24, 8, 25, 30, 10, 49, 42, 10, 73, 48, 12, 121, 60, 132, 145, 72, 134, 217, 84, 254, 265, 96, 376, 361, 114, 616, 433, 126, 858, 553, 864, 1218, 649, 882, 1580, 817, 1620, 2180, 937

Offset: 0

Ilya Gutkovskiy, Nov 22 2020

nonn

			a(15) = 6 because we have [8, 5, 2], [8, 2, 5], [5, 8, 2], [5, 2, 8], [2, 8, 5] and [2, 5, 8].

Index entries for sequences related to compositions

Cf. A000931, A016789, A032020, A032021, A035386, A262928, A337547, A339059, A339060.

nmax = 65; CoefficientList[Series[Sum[k! x^(k (3 k + 1)/2)/Product[1 - x^(3 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(3*k + 1)/2) / Product_{j=1..k} (1 - x^(3*j)).

A339059 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 4.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 4, 6, 0, 1, 4, 6, 0, 1, 6, 12, 0, 1, 6, 18, 24, 1, 8, 24, 24, 1, 8, 30, 48, 1, 10, 42, 72, 1, 10, 48, 120, 121, 12, 60, 144, 121, 12, 72, 216, 241, 14, 84, 264, 361, 14, 96, 360, 601, 16, 114, 432, 841, 736, 126, 552, 1201, 738

Offset: 0

Ilya Gutkovskiy, Nov 22 2020

nonn

			a(15) = 6 because we have [9, 5, 1], [9, 1, 5], [5, 9, 1], [5, 1, 9], [1, 9, 5] and [1, 5, 9].

Index entries for sequences related to compositions

Cf. A003269, A016813, A032020, A032021, A035451, A169975, A337547, A337548, A339060.

nmax = 70; CoefficientList[Series[Sum[k! x^(k (2 k - 1))/Product[1 - x^(4 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(2*k - 1)) / Product_{j=1..k} (1 - x^(4*j)).

A339060 Number of compositions (ordered partitions) of n into distinct parts congruent to 3 mod 4.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 4, 1, 0, 6, 4, 1, 0, 6, 6, 1, 0, 12, 6, 1, 0, 18, 8, 1, 24, 24, 8, 1, 24, 30, 10, 1, 48, 42, 10, 1, 72, 48, 12, 1, 120, 60, 12, 121, 144, 72, 14, 121, 216, 84, 14, 241, 264, 96, 16, 361, 360, 114, 16, 601, 432, 126, 18, 841

Offset: 0

Ilya Gutkovskiy, Nov 22 2020

nonn

			a(21) = 6 because we have [11, 7, 3], [11, 3, 7], [7, 11, 3], [7, 3, 11], [3, 11, 7] and [3, 7, 11].

Index entries for sequences related to compositions

Cf. A004767, A017817, A032020, A032021, A035462, A147599, A337547, A337548, A339059.

nmax = 75; CoefficientList[Series[Sum[k! x^(k (2 k + 1))/Product[1 - x^(4 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(2*k + 1)) / Product_{j=1..k} (1 - x^(4*j)).

A079499 Total number of parts in all partitions of n into distinct odd parts.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 2, 1, 4, 4, 4, 4, 6, 7, 6, 10, 12, 13, 12, 16, 18, 22, 22, 25, 32, 36, 36, 42, 50, 53, 58, 64, 76, 83, 88, 99, 116, 123, 132, 147, 168, 181, 194, 215, 240, 262, 280, 306, 346, 375, 396, 437, 482, 521, 558, 610, 670, 724, 772, 840, 922, 993, 1056, 1151, 1256, 1348

Offset: 0

Arnold Knopfmacher, Jan 21 2003

nonn

Also sum of the sizes of the Durfee squares of all self-conjugate partitions of n. Example: a(13)=7 because there are three self-conjugate partitions of 13, namely [7,1,1,1,1,1,1], [5,3,3,1,1] and [4,4,3,2], having Durfee squares of sizes 1,3 and 3, respectively. a(n) = Sum_{k=1..floor(sqrt(n))} k*A116422(n,k). - Emeric Deutsch, Feb 14 2006

			a(13)=7 because the partitions of 13 into distinct odd parts are [13], [9,3,1] and [7,5,1] and we have 1+3+3=7 parts.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000 (terms 0..1000 from T. D. Noe)
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Cf. A015723, A000700, A067619, A006128.
Cf. A032021.
Cf. A116422.

g:=sum(k*x^(k^2)/product(1-x^(2*i),i =1..k),k=1..20):gser:=series(g,x=0,52): seq(coeff(gser,x,n),n=0..50); # Emeric Deutsch, Feb 14 2006

max = 100; s = Sum[ k*x^(k^2) / Product[1-x^(2*j), {j, 1, k}], {k, 1, Sqrt[max] // Ceiling}]; CoefficientList[ Series[s, {x, 0, max}], x] (* Jean-François Alcover, Feb 19 2015, after Vladeta Jovovic *)

N=66;  S=2+sqrtint(N); x='x+O('x^N);
gf=sum(n=0, S, n*x^(n^2)/prod(k=1,n, 1-x^(2*k)) );
concat( [0], Vec(gf) )
\\ Joerg Arndt, Feb 18 2014

G.f.: (Sum_{k>=1} x^(2*k-1)/(1 + x^(2*k-1))) * Product_{m>=1} (1 + x^(2m-1)).
G.f.: Sum_{k>=1} k*x^(k^2)/Product_{j=1..k} (1 - x^(2*j)). - Vladeta Jovovic, Aug 06 2004
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/6)) / (Pi * 2^(5/4) * n^(1/4)). - Vaclav Kotesovec, May 20 2018

cp's OEIS Frontend

A219107 Number of compositions (ordered partitions) of n into distinct prime parts.

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A331844 Number of compositions (ordered partitions) of n into distinct squares.

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A331843 Number of compositions (ordered partitions) of n into distinct triangular numbers.

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A218396 Number of compositions of n into distinct (nonzero) Fibonacci numbers.

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A331845 Number of compositions (ordered partitions) of n into distinct cubes.

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A337547 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 3.

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A337548 Number of compositions (ordered partitions) of n into distinct parts congruent to 2 mod 3.

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A339059 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 4.

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