A209318 -id:A209318 - cp's OEIS frontend

A008289 Triangle read by rows: Q(n,m) = number of partitions of n into m distinct parts, n>=1, m>=1.

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 4, 3, 1, 4, 4, 1, 1, 5, 5, 1, 1, 5, 7, 2, 1, 6, 8, 3, 1, 6, 10, 5, 1, 7, 12, 6, 1, 1, 7, 14, 9, 1, 1, 8, 16, 11, 2, 1, 8, 19, 15, 3, 1, 9, 21, 18, 5, 1, 9, 24, 23, 7, 1, 10, 27, 27, 10, 1, 1, 10, 30, 34, 13, 1, 1, 11, 33, 39, 18, 2, 1, 11, 37

Offset: 1

N. J. A. Sloane

Row n contains A003056(n) = floor((sqrt(8*n+1)-1)/2) terms (number of terms increases by one at each triangular number). - Michael Somos, Dec 04 2002
Row sums give A000009.
Q(n,m) is the number of partitions of n whose greatest part is m and every number in {1,2,...,m} occurs as a part at least once. - Geoffrey Critzer, Nov 17 2011

			Q(8,3) = 2 since 8 can be written in 2 ways as sum of 3 distinct positive integers: 5+2+1 and 4+3+1.
Triangle starts:
  1;
  1;
  1,  1;
  1,  1;
  1,  2;
  1,  2,  1;
  1,  3,  1;
  1,  3,  2;
  1,  4,  3;
  1,  4,  4,  1;
  1,  5,  5,  1;
  1,  5,  7,  2;
  1,  6,  8,  3;
  1,  6, 10,  5;
  1,  7, 12,  6,  1;
  1,  7, 14,  9,  1;
  1,  8, 16, 11,  2;
  1,  8, 19, 15,  3;
  1,  9, 21, 18,  5;
  1,  9, 24, 23,  7;
  1, 10, 27, 27, 10,  1;
  1, 10, 30, 34, 13,  1;
  1, 11, 33, 39, 18,  2;
  1, 11, 37, 47, 23,  3;
  1, 12, 40, 54, 30,  5;
  1, 12, 44, 64, 37,  7;
  1, 13, 48, 72, 47, 11;
  1, 13, 52, 84, 57, 14, 1;
  1, 14, 56, 94, 70, 20, 1; ...
Q(8,3) = 2 because there are 2 partitions of 8 in which  1, 2 and 3 occur as a part at least once: (3,2,2,1), (3,2,1,1,1). - _Geoffrey Critzer_, Nov 17 2011

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.

Alois P. Heinz, Rows n = 1..500 of triangle, flattened (first 200 rows from T. D. Noe)
Carolyn Echter, Georg Maier, Juan-Diego Urbina, Caio Lewenkopf, and Klaus Richter, Many-body density of states of bosonic and fermionic gases: a combinatorial approach, arXiv:2409.08696 [cond-mat.quant-gas], 2024. See p. 10.
Eric Weisstein's World of Mathematics, Partition Function Q.

Cf. A030699, A104382, A117147, A209318.
Sum of n-th row is A000009(n). Sum(Q(n,k)*k, k>=1) = A015723(n).
A060016 is another version.
Cf. A032020.

g:=product(1+t*x^j,j=1..40): gser:=simplify(series(g,x=0,32)): P[0]:=1: for n from 1 to 30 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 25 do seq(coeff(P[n],t,j),j=1..floor((sqrt(8*n+1)-1)/2)) od; # yields sequence in triangular form; Emeric Deutsch, Feb 21 2006
# second Maple program:
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
T:= n-> subsop(1=NULL, b(n, n))[]:
seq(T(n), n=1..40);  # Alois P. Heinz, Nov 18 2012

q[n_, k_] := q[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]]; Take[ Flatten[ Table[q[n, k], {n, 1, 24}, {k, 1, Floor[(Sqrt[8n+1] - 1)/2]}]], 91] (* Jean-François Alcover, Aug 01 2011, after PARI prog. *)
(* As a triangular table: *)
Table[Coefficient[Series[Product[1+t    x^i,{i,n}],{x,0,n}],x^n t^m],{n,24},{m,n}] (* Wouter Meeussen, Feb 22 2014 *)
Table[Count[PowersRepresentations[n, k, 1], ?(Nor[MemberQ[#, 0], MemberQ[Differences@ #, 0]] &)], {n, 23}, {k, Floor[(Sqrt[8 n + 1] - 1)/2]}] // Flatten (* _Michael De Vlieger, Jul 12 2017 *)
nrows = 24; d=Table[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &],{n, nrows}] ;
Flatten@Table[Table[Count[d[[n]], x_ /; Length[x] == m], {m, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, nrows}] (* Robert Price, Aug 17 2020 *)

{Q(n, k) = if( k<0 || k>n,0, polcoeff( polcoeff( prod(i=1, n, 1 + y*x^i, 1 + x * O(x^n)), n), k))}; /* Michael Somos, Dec 04 2002 */

PARI
```
Q(n,k)=if(nPaul D. Hanna
```

{Q(n, k) = my(u); if( n<1 || k<1 || k>(sqrtint(8*n+1)-1)\2, 0, u = n - k *(k+1)/2; polcoeff( 1 / prod(i=1, k, 1 - x^i, 1 + x*O(x^u)), u))}; /* Michael Somos, Jul 11 2017 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A008289_T(n,k):
    if k<1 or nA008289_T(n-k,k)+A008289_T(n-k,k-1) # Chai Wah Wu, Sep 22 2023

G.f.: Product_{n>0} (1 + y*x^n) = 1 + Sum_{n>0, k>0} Q(n, k) * x^n * y^k. - Michael Somos, Dec 04 2002
Q(n, k) = Q(n-k, k) + Q(n-k, k-1) for n>k>=1, with Q(1, 1)=1, Q(n, 0)=0 (n>=1). - Paul D. Hanna, Mar 04 2005
G.f.: Sum_{n>0, k>0} x^n * y^(k*(k+1)/2) / Product_{i=1..k} (1 - y^i). - Michael Somos, Jul 11 2017
Sum_{k>=0} k! * Q(n,k) = A032020(n). - Alois P. Heinz, Feb 25 2020
Q(n, m) = A008284(n - m*(m-1)/2, m) = A026820(n - m*(m+1)/2, m), using for the latter, the extension A026820(n, k) = A026820(n, n) = A000041(n), for every k >= n >= 0. - Álvar Ibeas, Jul 23 2020

Additional comments from Michael Somos, Dec 04 2002

Entry revised by N. J. A. Sloane, Nov 20 2006

A117147 Triangle read by rows: T(n,k) is the number of partitions of n with k parts in which no part occurs more than 3 times (n>=1, k>=1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 3, 4, 3, 1, 1, 4, 5, 4, 2, 1, 4, 7, 6, 3, 1, 1, 5, 8, 9, 5, 1, 1, 5, 10, 11, 8, 3, 1, 6, 12, 14, 11, 5, 1, 1, 6, 14, 18, 15, 8, 2, 1, 7, 16, 23, 20, 11, 4, 1, 7, 19, 27, 27, 17, 6, 1, 1, 8, 21, 33, 34, 23, 10, 2, 1, 8, 24, 39, 43, 32, 15, 4, 1, 9

Offset: 1

Emeric Deutsch, Mar 07 2006

Row n has floor(sqrt(6n+6)-3/2) terms. Row sums yield A001935. Sum(k*T(n,k),k>=0) = A117148(n).

			T(7,3) = 4 because we have [5,1,1], [4,2,1], [3,3,1] and [3,2,2].
Triangle starts:
1;
1, 1;
1, 1, 1;
1, 2, 1;
1, 2, 2, 1;
1, 3, 3, 2;
1, 3, 4, 3, 1;

Alois P. Heinz, Rows n = 1..350, flattened

Cf. A001935, A008289, A117148, A209318.

g:=-1+product(1+t*x^j+t^2*x^(2*j)+t^3*x^(3*j),j=1..35): gser:=simplify(series(g,x=0,23)): for n from 1 to 18 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 18 do seq(coeff(P[n],t^j),j=1..floor(sqrt(6*n+6)-3/2)) od; # yields sequence in triangular form
# second Maple program
b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else []; for j from 0 to min(3, n/i) do
      zip((x, y)->x+y, %, [0$j, b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> subsop(1=NULL, [b(n, n)])[]:
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 08 2013

max = 18; g = -1+Product[1+t*x^j+t^2*x^(2j)+t^3*x^(3j), {j, 1, max}]; t[n_, k_] := SeriesCoefficient[g, {x, 0, n}, {t, 0, k}]; Table[DeleteCases[Table[t[n, k], {k, 1, n}], 0], {n, 1, max}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

G.f.: G(t,x) = -1+product(1+tx^j+t^2*x^(2j)+t^3*x^(3j), j=1..infinity).

A185350 Number of parts in all partitions of n in which no part occurs more than twice.

Original entry on oeis.org

0, 1, 3, 3, 8, 11, 17, 23, 36, 48, 69, 88, 125, 157, 212, 271, 356, 445, 574, 711, 906, 1118, 1400, 1711, 2125, 2583, 3171, 3828, 4666, 5604, 6777, 8095, 9730, 11567, 13815, 16357, 19429, 22910, 27077, 31801, 37432, 43802, 51338, 59871, 69914, 81273, 94562

Offset: 0

Alois P. Heinz, Jan 21 2013

nonn

			a(6) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
a(7) = 23: [7], [6,1], [5,2], [4,3], [5,1,1], [4,2,1], [3,3,1], [3,2,2], [3,2,1,1].

Vaclav Kotesovec, Table of n, a(n) for n = 0..11410 (terms 0..1000 from Alois P. Heinz)

Column k=2 of A210485.

Cf. A015723, A117148, A209318.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1)), j=0..min(n/i, 2))))
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);

b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[b[n - i j, i - 1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]] j}, {j, 0, Min[n/i, k]} ] ] ];
a[n_] := b[n, n, 2][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)
Table[Length[Flatten[Select[IntegerPartitions[n],Max[Length/@Split[#]]<3&]]],{n,0,50}] (* Harvey P. Dale, Jul 04 2023 *)

a(n) = Sum_{k>=0} k*A209318(n,k).

a(n) ~ log(3) * exp(2*Pi*sqrt(n)/3) / (2*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018

A320592 Number of partitions of n with four parts in which no part occurs more than twice.

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 12, 14, 19, 23, 29, 34, 42, 48, 58, 66, 77, 87, 101, 112, 128, 142, 160, 176, 197, 215, 239, 260, 286, 310, 340, 366, 399, 429, 465, 498, 538, 574, 618, 658, 705, 749, 801, 848, 904, 956, 1016, 1072, 1137, 1197, 1267, 1332, 1406, 1476, 1556

Offset: 6

Alois P. Heinz, Oct 16 2018

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Column k=4 of A209318.

G.f.: -(x^4-x^2-1)*x^6/((x^2+1)*(x^2+x+1)*(x+1)^2*(x-1)^4).

A320593 Number of partitions of n with five parts in which no part occurs more than twice.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 16, 21, 28, 36, 46, 57, 71, 86, 104, 124, 148, 173, 203, 235, 272, 312, 357, 405, 460, 518, 583, 652, 729, 810, 900, 995, 1099, 1209, 1329, 1455, 1593, 1737, 1893, 2057, 2234, 2419, 2618, 2826, 3049, 3282, 3531, 3790, 4067, 4355, 4661, 4980

Offset: 9

Alois P. Heinz, Oct 16 2018

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Column k=5 of A209318.

G.f.: (x^5-x-1)*x^9/((x^2+x+1)*(x^4+x^3+x^2+x+1)*(x^2+1)*(x+1)^2*(x-1)^5).

A320594 Number of partitions of n with six parts in which no part occurs more than twice.

Original entry on oeis.org

1, 1, 3, 5, 8, 11, 18, 23, 33, 44, 58, 73, 96, 117, 148, 181, 221, 264, 321, 377, 449, 526, 616, 712, 830, 949, 1093, 1246, 1420, 1605, 1822, 2044, 2302, 2575, 2880, 3202, 3569, 3947, 4375, 4826, 5322, 5844, 6427, 7029, 7698, 8400, 9164, 9965, 10846, 11757

Offset: 12

Alois P. Heinz, Oct 16 2018

Alois P. Heinz, Table of n, a(n) for n = 12..10000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -2, 0, 1, 1, 1, 1, 0, -2, 0, -1, 0, 0, 1, 1, -1).

Column k=6 of A209318.

LinearRecurrence[{1,1,0,0,-1,0,-2,0,1,1,1,1,0,-2,0,-1,0,0,1,1,-1},{1,1,3,5,8,11,18,23,33,44,58,73,96,117,148,181,221,264,321,377,449},50] (* Harvey P. Dale, Dec 19 2022 *)

G.f.: (x^9-x^8-x^7-x^5+x^3+x^2+1)*x^12 / ((x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(x^2+x+1)^2 *(x+1)^3 *(x-1)^6).

A320595 Number of partitions of n with seven parts in which no part occurs more than twice.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 21, 30, 41, 57, 75, 99, 129, 165, 207, 261, 322, 396, 483, 585, 701, 840, 995, 1176, 1382, 1617, 1880, 2183, 2518, 2898, 3322, 3797, 4321, 4911, 5556, 6275, 7066, 7939, 8892, 9947, 11092, 12350, 13719, 15214, 16832, 18601, 20507, 22580

Offset: 16

Alois P. Heinz, Oct 16 2018

Alois P. Heinz, Table of n, a(n) for n = 16..10000

Column k=7 of A209318.

G.f.: -(x^12-x^10-x^8-x^7+x^3+x+1)*x^16 / ((x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+1) *(x^2+x+1)^2 *(x+1)^3 *(x-1)^7).

A320596 Number of partitions of n with eight parts in which no part occurs more than twice.

Original entry on oeis.org

1, 1, 3, 5, 9, 13, 21, 29, 43, 59, 82, 108, 146, 187, 244, 309, 393, 488, 610, 746, 917, 1109, 1343, 1606, 1924, 2277, 2698, 3168, 3719, 4331, 5045, 5833, 6744, 7750, 8900, 10167, 11609, 13189, 14976, 16934, 19132, 21533, 24219, 27143, 30399, 33939, 37859

Offset: 20

Alois P. Heinz, Oct 16 2018

Alois P. Heinz, Table of n, a(n) for n = 20..10000

Column k=8 of A209318.

G.f.: (x^15-x^11-x^10-x^8-x^7+x^4+x^3+x^2+1)*x^20 / ((x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+1)^2 *(x^2+x+1)^2 *(x+1)^4 *(x-1)^8).

A320597 Number of partitions of n with nine parts in which no part occurs more than twice.

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 24, 35, 50, 71, 98, 131, 178, 234, 304, 393, 503, 633, 798, 992, 1225, 1506, 1839, 2226, 2692, 3229, 3856, 4587, 5434, 6400, 7526, 8804, 10266, 11934, 13830, 15964, 18397, 21123, 24192, 27640, 31505, 35805, 40629, 45978, 51928, 58531, 65849

Offset: 25

Alois P. Heinz, Oct 16 2018

Alois P. Heinz, Table of n, a(n) for n = 25..10000

Column k=9 of A209318.

Table[Count[IntegerPartitions[n,{9}],?(Max[Length/@Split[#]]<3&)],{n,25,75}] (* _Harvey P. Dale, Nov 10 2022 *)

G.f.: (x^20-x^19-x^18-x^16+x^14+2*x^11+x^10+x^9+x^8-x^6-x^4-x^3-x-1)*x^25 / ((x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^6+x^3+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+1)^2 *(x^2+x+1)^3 *(x+1)^4 *(x-1)^9).

A320598 Number of partitions of n with ten parts in which no part occurs more than twice.

Original entry on oeis.org

1, 1, 3, 5, 9, 14, 23, 32, 49, 69, 98, 133, 184, 242, 324, 421, 547, 698, 893, 1119, 1408, 1745, 2160, 2646, 3240, 3923, 4751, 5705, 6838, 8142, 9682, 11434, 13491, 15830, 18540, 21616, 25164, 29160, 33748, 38906, 44778, 51364, 58833, 67165, 76579, 87060

Offset: 30

Alois P. Heinz, Oct 16 2018

Alois P. Heinz, Table of n, a(n) for n = 30..10000

Column k=10 of A209318.

G.f.: -(x^25 -x^23 -x^21 -x^20 -x^17 +x^16 +x^14 +2*x^13 +x^12 +x^11 +x^10 +x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -1) *x^30 / ((x^2-x+1) *(x^4-x^3+x^2-x+1) *(x^6+x^3+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+1)^2 *(x^4+x^3+x^2+x+1)^2 *(x^2+x+1)^3 *(x+1)^5 *(x-1)^10).

cp's OEIS Frontend

A008289 Triangle read by rows: Q(n,m) = number of partitions of n into m distinct parts, n>=1, m>=1.

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A117147 Triangle read by rows: T(n,k) is the number of partitions of n with k parts in which no part occurs more than 3 times (n>=1, k>=1).

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A185350 Number of parts in all partitions of n in which no part occurs more than twice.

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Maple

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A320592 Number of partitions of n with four parts in which no part occurs more than twice.

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A320593 Number of partitions of n with five parts in which no part occurs more than twice.

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A320594 Number of partitions of n with six parts in which no part occurs more than twice.

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A320595 Number of partitions of n with seven parts in which no part occurs more than twice.

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A320596 Number of partitions of n with eight parts in which no part occurs more than twice.

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A320597 Number of partitions of n with nine parts in which no part occurs more than twice.