A001970 -id:A001970 - cp's OEIS frontend

A355742 Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.

1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 3, 0, 2, 0, 2, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Gus Wiseman, Jul 20 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 49 are {4,4}, and the a(49) = 4 choices are: (2,2), (2,4), (4,2), (4,4).
The prime indices of 777 are {2,4,12}, and the a(777) = 6 choices are: (2,2,2), (2,2,3), (2,2,4), (2,4,2), (2,4,3), (2,4,4).

Wikipedia, Cartesian product.

The unordered version is A001970, row-sums of A061260.
Positions of 1's are A076610, just primes A355743.
Positions of 0's are A299174.
Allowing all divisors (not just primes) gives A355731, firsts A355732.
Choosing only prime factors (not prime-powers) gives A355741.
Counting multisets of primes gives A355744.
The case of weakly increasing primes A355745, all divisors A355735.
A000688 counts factorizations into prime powers.
A001414 adds up distinct prime factors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
Cf. A000720, A120383, A279784, A289509, A324850, A355733, A355739, A355746.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@PrimeOmega/@primeMS[n],{n,100}]

Totally multiplicative with a(prime(k)) = A001222(k).

A318566 Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.

Original entry on oeis.org

1, 6, 21, 104, 452, 2335, 11992, 66810, 385101, 2336352, 14738380, 96831730, 659809115, 4657075074, 33974259046, 255781455848, 1984239830571, 15839628564349, 129951186405574, 1094486382191624, 9453318070371926, 83654146992936350, 757769011659766015, 7020652591448497490

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

			Non-isomorphic representatives of the a(3) = 21 multiset partitions of multiset partitions:
  {{{1,1,1}}}
  {{{1,1,2}}}
  {{{1,2,3}}}
  {{{1},{1,1}}}
  {{{1},{1,2}}}
  {{{1},{2,3}}}
  {{{2},{1,1}}}
  {{{1},{1},{1}}}
  {{{1},{1},{2}}}
  {{{1},{2},{3}}}
  {{{1}},{{1,1}}}
  {{{1}},{{1,2}}}
  {{{1}},{{2,3}}}
  {{{2}},{{1,1}}}
  {{{1}},{{1},{1}}}
  {{{1}},{{1},{2}}}
  {{{1}},{{2},{3}}}
  {{{2}},{{1},{1}}}
  {{{1}},{{1}},{{1}}}
  {{{1}},{{1}},{{2}}}
  {{{1}},{{2}},{{3}}}

Cf. A001970, A007716, A050336, A050338, A255906, A269134, A317533, A317791, A318564, A318565.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
dubnorm[m_]:=First[Union[Table[Map[Sort,m/.Rule@@@Table[{Union[Flatten[m]][[i]],Union[Flatten[m]][[perm[[i]]]]},{i,Length[perm]}],{0,2}],{perm,Permutations[Union[Flatten[m]]]}]]];
Table[Length[Union[dubnorm/@Join@@mps/@Join@@mps/@strnorm[n]]],{n,5}]

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=sExp(symGroupSeries(n))); NumUnlabeledObjsSeq(sCartProd(A, sExp(A)-1))} \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020

A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 6, 5, 1, 0, 1, 1, 5, 10, 14, 7, 1, 0, 1, 1, 6, 15, 30, 27, 11, 1, 0, 1, 1, 7, 21, 55, 75, 58, 15, 1, 0, 1, 1, 8, 28, 91, 170, 206, 111, 22, 1, 0, 1, 1, 9, 36, 140, 336, 571, 518, 223, 30, 1, 0

Offset: 0

Alois P. Heinz, Jul 28 2017

A(n,k) is the number of unlabeled rooted trees with exactly n leaves, all in level k.  A(3,3) = 6:
:    o        o       o        o        o       o
:    |        |       |       / \      / \     /|\
:    o        o       o      o   o    o   o   o o o
:    |       / \     /|\     |   |   ( )  |   | | |
:    o      o   o   o o o    o   o   o o  o   o o o
:   /|\    ( )  |   | | |   ( )  |   | |  |   | | |
:  o o o   o o  o   o o o   o o  o   o o  o   o o o

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  0, 1,  2,   3,    4,    5,     6,     7,      8, ...
  0, 1,  3,   6,   10,   15,    21,    28,     36, ...
  0, 1,  5,  14,   30,   55,    91,   140,    204, ...
  0, 1,  7,  27,   75,  170,   336,   602,   1002, ...
  0, 1, 11,  58,  206,  571,  1337,  2772,   5244, ...
  0, 1, 15, 111,  518, 1789,  5026, 12166,  26328, ...
  0, 1, 22, 223, 1344, 5727, 19193, 54046, 133476, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
Index entries for sequences related to rooted trees

Columns k=1-10 give: A000012, A000041, A001970, A007713, A007714, A290355, A290356, A290357, A290358, A290359.
Rows 0+1,2-10 give: A000012, A001477, A000217, A000330, A007715, A290360, A290361, A290362, A290363, A290364.
Main diagonal gives A290354.
Cf. A144150.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(A(d, k-1)*d, d=divisors(j))*A(n-j, k), j=1..n)/n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[A[d, k - 1]*d, {d, Divisors[j]}] A[n - j, k], {j, n}]/n]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}]//Flatten (* Indranil Ghosh, Jul 30 2017, after Maple code *)

G.f. of column k=0: 1+x, of column k>0: Product_{j>0} 1/(1-x^j)^A(j,k-1).

A317715 Number of ways to split an integer partition of n into consecutive subsequences with equal sums.

Original entry on oeis.org

1, 1, 3, 4, 9, 8, 21, 16, 39, 38, 64, 57, 146, 102, 186, 211, 352, 298, 593, 491, 906, 880, 1273, 1256, 2444, 1998, 3038, 3277, 4861, 4566, 7710, 6843, 10841, 10742, 14966, 15071, 24499, 21638, 31334, 32706, 47157, 44584, 67464, 63262, 91351, 94247, 125248

Offset: 0

Gus Wiseman, Sep 29 2018

nonn

			The a(4) = 9 constant-sum split partitions:
  (4),
  (31),
  (22), (2)(2),
  (211), (2)(11),
  (1111), (11)(11), (1)(1)(1)(1).
The a(6) = 21 constant-sum split partitions:
  (6),
  (51),
  (42),
  (411),
  (33), (3)(3),
  (321), (3)(21),
  (3111), (3)(111),
  (222), (2)(2)(2),
  (2211), (2)(2)(11),
  (21111), (21)(111), (2)(11)(11),
  (111111), (111)(111), (11)(11)(11), (1)(1)(1)(1)(1)(1).

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..500
Gus Wiseman, The a(8) = 39 constant-sum split partitions.
Gus Wiseman, The a(10) = 64 constant-sum split partitions.
Gus Wiseman, The a(12) = 146 constant-sum split partitions.

Cf. A000041, A001970, A063834, A316223, A317545, A317546, A319001.

Cf. A316245, A317508, A318434, A318683, A318684, A319794.

comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
Table[Sum[Length[Select[comps[y],SameQ@@Total/@#&]],{y,IntegerPartitions[n]}],{n,10}]

a(16)-a(46) from Hiroaki Yamanouchi, Oct 02 2018

A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 2, 8, 16, 48, 144, 352, 896, 2432, 7168, 16896, 46080, 114688, 303104, 843776, 2080768, 5308416, 13762560, 34865152, 87818240, 241172480, 583008256, 1503657984, 3762290688, 9604956160, 23689428992, 60532195328, 156397207552, 385137770496, 967978254336

Offset: 0

Gus Wiseman, Jul 09 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(4) = 16 splits:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (1),(2)    (1,1,2)
                  (2),(1)    (1,2,1)
                  (1),(1,1)  (1),(3)
                  (1,1),(1)  (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1),(1,2)
                             (1),(2,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1),(1,1,1)
                             (1,1,1),(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A074854.
Starting with a strict composition gives A336128.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(n) = Sum_{k=0..n} 2^(n-k) k! A008289(n,k).

A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 6, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 8, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 3, 2, 1, 8, 2, 2, 2, 4, 1, 8, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1, 5, 1, 4, 5

Offset: 1

Gus Wiseman, May 26 2018

nonn

			The a(60) = 8 factorizations are (2*2*3*5), (2*2*15), (3*4*5), (3*20), (4*15), (5*12), (6*10), (60). Missing from this list are (2*3*10), (2*5*6), (2*30).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001055, A001970, A007716, A034444, A045778, A259936, A281116, A285572, A302242, A303386, A303431, A305001, A305148, A305150.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,100}]

pairwise_indivisible(v) = { for(i=1,#v,for(j=i+1,#v,if(!(v[j]%v[i]),return(0)))); (1); };
A305149(n, m=n, facs=List([])) = if(1==n, pairwise_indivisible(Set(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A305149(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

More terms from Antti Karttunen, Oct 08 2018

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 37, 57, 89, 265, 309, 521, 745, 1129, 3005, 3545, 5685, 8201, 12265, 16629, 41369, 48109, 77265, 107645, 160681, 214861, 316913, 644837, 798861, 1207445, 1694269, 2437689, 3326705, 4710397, 6270513, 12246521, 14853625, 22244569, 30308033, 43706705, 57926577, 82166105, 107873221, 148081785, 257989961, 320873065, 458994657, 628016225, 875485585, 1165065733

Offset: 0

Gus Wiseman, Jul 10 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 5 splits:
  ()  (1)  (2)  (3)     (4)     (5)
                (12)    (13)    (14)
                (21)    (31)    (23)
                (1)(2)  (1)(3)  (32)
                (2)(1)  (3)(1)  (41)
                                (1)(4)
                                (2)(3)
                                (3)(2)
                                (4)(1)
The a(6) = 29 splits:
  (6)    (1)(5)   (1)(2)(3)
  (15)   (2)(4)   (1)(3)(2)
  (24)   (4)(2)   (2)(1)(3)
  (42)   (5)(1)   (2)(3)(1)
  (51)   (1)(23)  (3)(1)(2)
  (123)  (1)(32)  (3)(2)(1)
  (132)  (13)(2)
  (213)  (2)(13)
  (231)  (2)(31)
  (312)  (23)(1)
  (321)  (31)(2)
         (32)(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A336130.
Starting with a non-strict composition gives A336127.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Set partitions with distinct block-sums are A275780.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(50) from Max Alekseyev, Feb 14 2024

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183, 1260, 1352, 1436, 1536, 1628, 1736, 1836, 1953, 2061, 2187, 2304, 2439

Offset: 0

N. J. A. Sloane, Simon Plouffe, Stephen P. Humphries

Number of 2 X 2 square partitions of n.
1/((1-x^2)*(1-x^4)^2*(1-x^6)) is the Molien series for 4-dimensional representation of a certain group of order 192 [Nebe, Rains, Sloane, Chap. 7].
Number of ways of writing n as n = p+q+r+s so that p >= q, p >= r, q >= s, r >= s with p, q, r, s >= 1. That is, we can partition n as
pq
rs
with p >= q, p >= r, q >= s, r >= s.
The coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) is a(n+4), where s(n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding to the two row partition and * represents the inner or Kronecker product of symmetric functions. - Mike Zabrocki, Dec 22 2005
Let F() be the Fibonacci sequence A000045. Let f([x, y, z, w]) = F(x) * F(y) * F(z) * F(w). Let N([x, y, z, w]) = x^2 + y^2 + z^2 + w^2. Let Q(k) = set of all ordered quadruples of integers [x, y, z, w] such that 1 <= x <= y <= z <= w and N([x, y, z, w]) = k. Let P(n) = set of all unordered triples {q1, q2, q3} of elements of some Q(k) such that max(w1, w2, w3) = n and f(q1) + f(q2) = f(q3). Then a(n-1) is the number of elements of P(n). - Michael Somos, Jan 21 2015
Number of partitions of 2n+2 into 4 parts with alternating parity from smallest to largest (or vice versa). - Wesley Ivan Hurt, Jan 19 2021

			a(7) = 4:
41 32 31 22
11 11 21 21
G.f. = x^4 + x^5 + 3*x^6 + 4*x^7 + 7*x^8 + 9*x^9 + 14*x^10 + 17*x^11 + ...
a(5-1) = 1 because P(5) has only one triple {[1,1,1,5], [2,2,2,4], [1,3,3,3]} of elements from Q(28) where f([1,1,1,5]) = 5, f([2,2,2,4]) = 3, f([1,3,3,3]) = 8, and 5 + 3 = 8. - _Michael Somos_, Jan 21 2015
a(6-1) = 1 because P(6) has only one triple {[1,1,2,6], [2,2,3,5], [1,3,4,4]} of elements from Q(42) where f([1,1,2,6]) = 8, f([2,2,3,5]) = 10, f([1,3,4,4]) = 18 and 8 + 10 = 18. - _Michael Somos_, Jan 21 2015
a(7-1) = 3 because P(7) has three triples. The triple {[1,1,1,7], [2,4,4,4], [3,3,3,5]} from Q(52) where f([1,1,1,7]) = 13, f([2,4,4,4]) = 27, f([3,3,3,5]) = 40 and 13 + 27 = 40. The triple {[1,2,2,7], [2,3,3,6], [1,4,4,5]} from Q(58) where f([1,2,2,7]) = 13, f([2,3,3,6]) = 32, f([1,4,4,5]) = 45 and 13 + 32 = 45. The triple {[1,1,3,7], [2,2,4,6], [1,3,5,5]} from Q(60) where f([1,1,3,7]) = 26, f([2,2,4,6]) = 24, f([1,3,5,5]) = 50 and 26 + 24 = 50. - _Michael Somos_, Jan 21 2015

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242 (Cor. 2.1, n=1).
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 5.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2. [MR1219862 (94d:11029)]
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
S. P. Humphries, Home page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 232
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Michael Somos, In the Elliptic Realm
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

See A266769 for a version without the four leading zeros.
Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.
First differences of A097701.
Cf. A082424, A082437.

a:=[0,0,0,0,1,1,3,4];; for n in [9..60] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]-2*a[n-4]-a[n-5]+2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 10 2019

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(H);

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( x^4/((1-x)*(1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 10 2019

a:= n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,2,-1,-2,-1,2,1,-1][i] else 0 fi)^n)[1,5]: seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[x^4/((1-x)*(1-x^2)^2*(1-x^3)), {x,0,60}], x] (* Jean-François Alcover, Mar 30 2011 *)
LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{0,0,0,0,1,1,3,4},60] (* Harvey P. Dale, Mar 04 2012 *)
a[ n_]:= Quotient[9(n+1)(-1)^n +2n^3 -9n +65, 144]; (* Michael Somos, Jan 21 2015 *)
a[ n_]:= Sign[n] SeriesCoefficient[ x^4/((1-x)(1-x^2)^2(1-x^3)), {x, 0, Abs@n}]; (* Michael Somos, Jan 21 2015 *)

{a(n) = (9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65) \ 144}; /* Michael Somos, Jan 21 2015 */

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,1,2,-1,-2,-1,2,1]^n*[0;0;0;0;1;1;3;4])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

def AA008763_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4/((1-x)*(1-x^2)^2*(1-x^3))).list()
AA008763_list(60) # G. C. Greubel, Sep 10 2019

Let f4(n) = number of partitions n = p+q+r+s into exactly 4 parts, with p >= q >= r >= s >= 1 (see A026810, A001400) and let g4(n) be the number with q > r (so that g4(n) = f4(n-2)). Then a(n) = f4(n) + g4(n).
a(n) = (1/144)*( 2*n^3 + 9*n*((-1)^n - 1) - 16*((n is 2 mod 3) - (n is 1 mod 3)) ).
a(n) = (1/72)*(n+3)*(n+2)*(n+1)-(1/12)*(n+2)*(n+1)+(5/144)*(n+1)+(1/16)*(n+1)*(-1)^n+(1/16)*(-1)^(n+1)+(7/144)+(2*sqrt(3)/27)*sin(2*Pi*n/3). - Richard Choulet, Nov 27 2008
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8), n>7. - Harvey P. Dale, Mar 04 2012
a(n) = floor((9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65)/144). - Tani Akinari, Nov 06 2012
a(n+1) - a(n) = A008731(n-3). - R. J. Mathar, Aug 06 2013
a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 21 2015
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017

Entry revised Dec 25 2003

A055884 Euler transform of partition triangle A008284.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 4, 5, 1, 4, 8, 7, 7, 1, 6, 12, 16, 12, 11, 1, 6, 17, 25, 28, 19, 15, 1, 8, 22, 43, 49, 48, 30, 22, 1, 8, 30, 58, 87, 88, 77, 45, 30, 1, 10, 36, 87, 134, 167, 151, 122, 67, 42, 1, 10, 45, 113, 207, 270, 296, 247, 185, 97, 56, 1, 12, 54, 155, 295, 448, 510, 507, 394, 278, 139, 77

Offset: 1

Christian G. Bower, Jun 09 2000

Number of multiset partitions of length-k integer partitions of n. - Gus Wiseman, Nov 09 2018

			From _Gus Wiseman_, Nov 09 2018: (Start)
Triangle begins:
   1
   1   2
   1   2   3
   1   4   4   5
   1   4   8   7   7
   1   6  12  16  12  11
   1   6  17  25  28  19  15
   1   8  22  43  49  48  30  22
   1   8  30  58  87  88  77  45  30
   ...
The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:
  {{5}}   {{1,4}}     {{1,1,3}}       {{1,1,1,2}}         {{1,1,1,1,1}}
          {{2,3}}     {{1,2,2}}      {{1},{1,1,2}}       {{1},{1,1,1,1}}
         {{1},{4}}   {{1},{1,3}}     {{1,1},{1,2}}       {{1,1},{1,1,1}}
         {{2},{3}}   {{1},{2,2}}     {{2},{1,1,1}}      {{1},{1},{1,1,1}}
                     {{2},{1,2}}    {{1},{1},{1,2}}     {{1},{1,1},{1,1}}
                     {{3},{1,1}}    {{1},{2},{1,1}}    {{1},{1},{1},{1,1}}
                    {{1},{1},{3}}  {{1},{1},{1},{2}}  {{1},{1},{1},{1},{1}}
                    {{1},{2},{2}}
(End)

Alois P. Heinz, Rows n = 1..200, flattened
N. J. A. Sloane, Transforms

Row sums give A001970.
Main diagonal gives A000041.
Columns k=1-2 give: A057427, A052928.
T(n+2,n+1) gives A000070.
T(2n,n) gives A360468.
Cf. A055885, A055886, A360742.
Cf. A000219, A007716, A008284, A255906, A317449, A317532, A317533, A320796, A320801, A320808.

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Feb 17 2023

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]]]];
Table[Length[Join@@mps/@IntegerPartitions[n,{k}]],{n,5},{k,n}] (* Gus Wiseman, Nov 09 2018 *)

cp's OEIS Frontend

A319719 Number of non-isomorphic connected antichains of multisets of weight n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A355742 Number of ways to choose a sequence of prime-power divisors, one of each prime index of n. Product of bigomega over the prime indices of n, with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A318566 Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A317715 Number of ways to split an integer partition of n into consecutive subsequences with equal sums.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

Views

Author

Keywords

Comments

Examples

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).