A007042 -id:A007042 - cp's OEIS frontend

A128568 Column 1 of triangle A128567.

1, 6, 31, 133, 587, 2531, 10950, 47185, 203704, 879711, 3804530, 16464710, 71312805, 309083291, 1340546867, 5817555402, 25258769216, 109711224970, 476675868834, 2071569641859, 9004430215111, 39144480326143, 170184867215647, 739924236443359, 3217001700174226

Offset: 0

Paul D. Hanna, Mar 12 2007

nonn

A128567 is the matrix square of Parker's partition triangle A047812.

Cf. A007042, A047812, A128567, A128569 (column 2), A128602 (row sums).

{a(n)=local(M);M=matrix(n+2,n+2,r,c,if(r

a(n) = Sum_{s=1..n+1} A047812(n+2,s)*A047812(s+1,1) = Sum_{s=1..n+1} A047812(n+2,s)*A007042(s+1) for n >= 0. - Petros Hadjicostas, May 31 2020

A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 9, 28, 36, 16, 1, 13, 69, 160, 164, 61, 1, 20, 160, 580, 1022, 855, 272, 1, 28, 337, 1837, 4996, 7072, 4988, 1385

Offset: 2

Gus Wiseman, Jan 03 2020

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			Triangle begins:
    1
    1    1
    1    3    2
    1    5    9    5
    1    9   28   36   16
    1   13   69  160  164   61
    1   20  160  580 1022  855  272
    1   28  337 1837 4996 7072 4988 1385
Row n = 5 counts the following multisystems (strings of 1's are replaced by their lengths):
  5  {1,4}      {{1},{1,3}}      {{{1}},{{1},{1,2}}}
     {2,3}      {{1},{2,2}}      {{{1,1}},{{1},{2}}}
     {1,1,3}    {{2},{1,2}}      {{{1}},{{2},{1,1}}}
     {1,2,2}    {{3},{1,1}}      {{{1,2}},{{1},{1}}}
     {1,1,1,2}  {{1},{1,1,2}}    {{{2}},{{1},{1,1}}}
                {{1,1},{1,2}}
                {{2},{1,1,1}}
                {{1},{1},{1,2}}
                {{1},{2},{1,1}}

Row sums are A318813.
Column k = 3 is A007042.
Column k = 4 is A001970(n) - 3*A000041(n) + 3.
Column k = n is A000111.
Row n is row prime(n) of A330727.
Cf. A000669, A001055, A002846, A005121, A196545, A213427, A318812, A320160, A330474, A330475, A330655, A330667, A330679.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

T(n,3) = A000041(n) - 2.

T(n,4) = A001970(n) - 3 * A000041(n) + 3.

A330936 Number of nontrivial factorizations of n into factors > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 1, 2, 0, 3, 0, 5, 0, 0, 0, 7, 0, 0, 0, 5, 0, 3, 0, 2, 2, 0, 0, 10, 0, 2, 0, 2, 0, 5, 0, 5, 0, 0, 0, 9, 0, 0, 2, 9, 0, 3, 0, 2, 0, 3, 0, 14, 0, 0, 2, 2, 0, 3, 0, 10, 3, 0, 0, 9, 0, 0

Offset: 1

Gus Wiseman, Jan 04 2020

nonn

The trivial factorizations of a number are (1) the case with only one factor, and (2) the factorization into prime numbers.

			The a(n) nontrivial factorizations of n = 8, 12, 16, 24, 36, 48, 60, 72:
  (2*4)  (2*6)  (2*8)    (3*8)    (4*9)    (6*8)      (2*30)    (8*9)
         (3*4)  (4*4)    (4*6)    (6*6)    (2*24)     (3*20)    (2*36)
                (2*2*4)  (2*12)   (2*18)   (3*16)     (4*15)    (3*24)
                         (2*2*6)  (3*12)   (4*12)     (5*12)    (4*18)
                         (2*3*4)  (2*2*9)  (2*3*8)    (6*10)    (6*12)
                                  (2*3*6)  (2*4*6)    (2*5*6)   (2*4*9)
                                  (3*3*4)  (3*4*4)    (3*4*5)   (2*6*6)
                                           (2*2*12)   (2*2*15)  (3*3*8)
                                           (2*2*2*6)  (2*3*10)  (3*4*6)
                                           (2*2*3*4)            (2*2*18)
                                                                (2*3*12)
                                                                (2*2*2*9)
                                                                (2*2*3*6)
                                                                (2*3*3*4)

Positions of nonzero terms are A033942.
Positions of 1's are A030078.
Positions of 2's are A054753.
Nontrivial integer partitions are A007042.
Nontrivial set partitions are A008827.
Nontrivial divisors are A070824.
Cf. A001055, A003238, A005121, A317145, A317176, A318812, A330665, A330935.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[DeleteCases[Rest[facs[n]],{_}]],{n,100}]

For prime n, a(n) = 0; for nonprime n, a(n) = A001055(n) - 2.

A058400 Triangle of partial row sums of partition triangle A058398.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 7, 6, 5, 3, 1, 11, 10, 9, 7, 4, 1, 15, 14, 13, 11, 8, 4, 1, 22, 21, 20, 18, 15, 10, 5, 1, 30, 29, 28, 26, 23, 18, 12, 5, 1, 42, 41, 40, 38, 35, 30, 23, 14, 6, 1, 56, 55, 54, 52, 49, 44, 37, 27, 16, 6, 1, 77, 76, 75, 73, 70, 65, 58, 47, 34, 19, 7, 1, 101

Offset: 1

Wolfdieter Lang, Dec 11 2000

Mirror of A026820. - Omar E. Pol, Apr 21 2012

			Triangle begins:
1;
2,   1;
3,   2,  1;
5,   4,  3,  1;
7,   6,  5,  3, 1;
11, 10,  9,  7, 4, 1;
15, 14, 13, 11, 8, 4, 1;

Columns 1-3: A000041(n), A000065(n), A007042(n+1).

Cf. A008284.

a(n, m) = sum(A058398(n, k), k=m..n).

A137614 Triangle read by rows: A000012 * A047812 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 8, 1, 5, 18, 28, 12, 1, 6, 31, 76, 63, 19, 1, 7, 51, 176, 232, 131, 27, 1, 8, 79, 370, 693, 617, 248, 39, 1, 9, 119, 722, 1821, 2284, 1458, 450, 53, 1, 10, 173, 1337, 4338, 7243, 6553, 3211, 773, 74, 1

Offset: 0

Gary W. Adamson, Jan 30 2008

Row sums = A014138: (1, 3, 8, 22, 64, 196, 625, ...).
From Petros Hadjicostas, Jun 01 2020: (Start)
We prove the claim above. From Guy (1992, 1993), we know that A000108(n) = Sum_{k=0..n-1} A047812(k) (the row sums of Parker's triangle are Catalan numbers).
We then have Sum_{k=0..n-1} T(n,k) = Sum_{k=0..n-1} Sum_{s=k+1..n} A047812(s,k) = Sum_{s=1..n} Sum_{k=0..s-1} A047812(s,k) = Sum_{s=1..n} A000108(s) = A014138(n) because A014138 contains partial sums of the Catalan numbers. (End)

			Triangle T(n,k) (with rows n >= 1 and columns k = 0..n-1) begins:
  1;
  2,  1;
  3,  4,   1;
  4,  9,   8,   1;
  5, 18,  28,  12,   1;
  6, 31,  76,  63,  19,  1;
  7, 51, 176, 232, 131, 27, 1;
  ...

R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A000012, A014138, A047812.

A(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
T(n,k) = sum(s=k+1, n, A(s,k));
vector(15, n, vector(n, k, T(n, k-1))) \\ Petros Hadjicostas, Jun 01 2020

T(n,k) = Sum_{s=k+1..n} A047812(s,k) for n >= 1 and 0 <= k <= n-1. - Petros Hadjicostas, Jun 01 2020

A330785 Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 8, 4, 0, 1, 9, 25, 28, 11, 0, 1, 13, 57, 111, 99, 33, 0, 1, 20, 129, 379, 561, 408, 116, 0, 1, 28, 253, 1057, 2332, 2805, 1739, 435, 0, 1, 40, 496, 2833, 8695, 15271, 15373, 8253, 1832, 0, 1, 54, 898, 6824, 28071, 67790, 98946, 85870, 40789, 8167

Offset: 1

Gus Wiseman, Jan 03 2020

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   3   2
   0   1   5   8   4
   0   1   9  25  28  11
   0   1  13  57 111  99  33
   0   1  20 129 379 561 408 116
Row n = 5 counts the following chains (minimum and maximum not shown):
  ()  (14)    (113)->(14)    (1112)->(113)->(14)
      (23)    (113)->(23)    (1112)->(113)->(23)
      (113)   (122)->(14)    (1112)->(122)->(14)
      (122)   (122)->(23)    (1112)->(122)->(23)
      (1112)  (1112)->(14)
              (1112)->(23)
              (1112)->(113)
              (1112)->(122)

Row sums are A213427.
Main diagonal is A002846.
Column k=3 is A007042.
Dominated by A330784.
The version for set partitions is A008826.
The version for factorizations is A330935.
Cf. A000111, A000258, A000311, A005121, A141268, A196545, A265947, A300383, A306186, A317141, A317176, A318813, A320160, A330679.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
upr[q_]:=Union[Sort/@Apply[Plus,mps[q],{2}]];
paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y,#}&/@DeleteCases[upr[y],y],{y,Sort/@IntegerPartitions[n]}],ConstantArray[1,n],{n}],Length[#]==k-1&]],{n,8},{k,n}]

T(n,k) = A330935(2^n,k).

A335323 First lower diagonal of Parker's triangle A047812.

Original entry on oeis.org

0, 1, 3, 7, 11, 18, 26, 38, 52, 73, 97, 131, 172, 227, 293, 381, 486, 623, 788, 998, 1251, 1571, 1954, 2432, 3006, 3714, 4561, 5600, 6838, 8345, 10139, 12306, 14879, 17973, 21633, 26011, 31181, 37334, 44579, 53170, 63257, 75171, 89130, 105554, 124750, 147269

Offset: 1

Petros Hadjicostas, May 31 2020

nonn

Apparently, this sequence was originally intended to be A7043 (now A007043), but for some reason it was crossed out on p. 4 of the annotated copy of Guy's 1992 preprint.

a(n) is the number of partitions of (n-2)*(n+1) into at most n parts each no bigger than n. Thus, a(n) is the coefficient of q^((n-2)*(n+1)) in the q-binomial coefficient [2*n, n].

			a(1) = 0 because it does not make sense to talk about the partitions of (1-2)*(1+1) = -2.
a(2) = 1 because we have only the empty partition for (2-2)*(2+1) = 0.
a(3) = 3 because we have the following partitions of (3-2)*(3+1) = 4 into no more than 3 parts each no bigger than 3: 1+3 = 1+1+2 = 2+2.
a(4) = 7 because we have the following partitions of (4-2)*(4+1) = 10 into no more than 4 parts each no bigger than 4: 2+4+4 = 3+3+4 = 1+1+4+4 = 1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3.
The PARI function partitions((n-2)*(n+1), n, n) can generate these partitions.

Alois P. Heinz, Table of n, a(n) for n = 1..500
R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A007042, A007043, A007044, A007045, A047812, A051643.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i b((n-2)*(n+1), n$2):
seq(a(n), n=1..50);  # Alois P. Heinz, May 31 2020

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
a[n_] := b[(n-2)(n+1), n, n];
Array[a, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
for(n=1, 43, print1(T(n, n-2), ", "))

A376348 a(n) is the number of multisets with n primes with which an n-gon with perimeter prime(n) can be formed.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 7, 7, 12, 19, 19, 25, 44, 72, 72, 119, 147, 152, 234, 292, 435, 777, 920, 946, 1135, 1161, 1377, 3702, 4293, 5942, 5942, 10741, 10741, 14483, 18953, 22091, 28658, 37686, 37686, 63053, 63053, 72389, 72389, 132732, 233773, 265312, 265312, 300443, 373266

Offset: 3

Felix Huber, Oct 13 2024

nonn

a(n) is the number of partitions of prime(n) into n prime parts < prime(n)/2.

First differs from A259254 at n=31: a(31) = 3702 but A259254(31) = 3703.

			a(7) = 2 because exactly the 2 partitions (2, 2, 2, 2, 3, 3, 3) and (2, 2, 2, 2, 2, 2, 5) have 7 prime parts and their sum is p(7) = 17.

Eric Weisstein's World of Mathematics, Polygon

Cf. A005044, A007042, A038499, A062890, A069906, A069907, A259254, A288253, A288254, A288255, A288256, A318604, A366398.

A376348:=proc(n)
   local a,p,x,i;
   a:=0;
   p:=ithprime(n);
   for x from NumberTheory:-pi(p/n)+1 to NumberTheory:-pi(p/2) do
      a:=a+numelems(select(i->nops(i)=n-1 and andmap(isprime,i),combinat:-partition(ithprime(n)-ithprime(x),ithprime(x))))
   od;
   return a
end proc;
seq(A376348(n),n=3..42);

a(n)={my(m=prime(n), p=primes(primepi((m-1)\2))); polcoef(polcoef(1/prod(i=1, #p, 1 - y*x^p[i], 1 + O(x*x^m)), m),n)} \\ Andrew Howroyd, Oct 13 2024

a(43) onwards from Andrew Howroyd, Oct 13 2024

A367106 Triangle read by rows where T(n,k) is the number of complete length-k integer partitions of n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 1, 3, 2, 1, 1, 0, 0, 0, 0, 3, 3, 2, 1, 1, 0, 0, 0, 0, 4, 5, 3, 2, 1, 1, 0, 0, 0, 0, 3, 5, 5, 3, 2, 1, 1, 0, 0, 0, 0, 4, 8, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 2, 9, 9, 7, 5

Offset: 0

Gus Wiseman, Nov 09 2023

An integer partition of n is complete (ranks A325781) if every integer from 0 to n is the sum of some submultiset of the parts.

			Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  0  1  1
  0  0  0  2  1  1
  0  0  0  1  2  1  1
  0  0  0  1  3  2  1  1
  0  0  0  0  3  3  2  1  1
  0  0  0  0  4  5  3  2  1  1
  0  0  0  0  3  5  5  3  2  1  1
  0  0  0  0  4  8  7  5  3  2  1  1
  0  0  0  0  2  9  9  7  5  3  2  1  1
  0  0  0  0  2 11 12 11  7  5  3  2  1  1
  0  0  0  0  1 11 16 13 11  7  5  3  2  1  1
  0  0  0  0  1 14 21 19 15 11  7  5  3  2  1  1
Row n = 11 counts the following partitions (empty columns not shown):
  6311  62111  611111  5111111  41111111  311111111  2111111111  11111111111
  6221  53111  521111  4211111  32111111  221111111
  5321  52211  431111  3311111  22211111
  4421  44111  422111  3221111
        43211  332111  2222111
        42221  322211
        33311  222221
        33221

Column k appears to have A000325(k) nonzero terms.
Column sums are A003513.
Central column T(2n,n) is A007042.
Row sums are A126796, ranks A325781.
The strict case is too sparse, row sums A188431 (complement A365831).
Grouping by maximum instead of length gives A261036.
A000041 counts integer partitions.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts subset-sums of prime indices, firsts A259941.
A365924 counts incomplete partitions, ranks A365830.
Cf. A002033, A018818, A046663, A047967, A055932, A276024, A304792, A365921.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n,{k}],nmz[#]=={}&]],{n,0,15},{k,0,n}]

cp's OEIS Frontend

A128568 Column 1 of triangle A128567.

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A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

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A330936 Number of nontrivial factorizations of n into factors > 1.

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A058400 Triangle of partial row sums of partition triangle A058398.

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A137614 Triangle read by rows: A000012 * A047812 as infinite lower triangular matrices.

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A330785 Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.

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A335323 First lower diagonal of Parker's triangle A047812.

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Maple

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A376348 a(n) is the number of multisets with n primes with which an n-gon with perimeter prime(n) can be formed.

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