A114921 -id:A114921 - cp's OEIS frontend

A226541 Number of unimodal compositions of n where the maximal part appears three times.

1, 0, 0, 1, 0, 0, 1, 2, 3, 5, 7, 11, 16, 24, 34, 51, 71, 102, 143, 201, 276, 384, 522, 714, 964, 1301, 1739, 2328, 3084, 4085, 5377, 7064, 9226, 12036, 15616, 20228, 26092, 33584, 43067, 55125, 70308, 89502, 113598, 143889, 181755, 229160, 288186, 361750, 453046, 566346, 706464

Offset: 0

Joerg Arndt, Jun 10 2013

nonn

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

Cf. A006330 (max part appears once), A114921 (max part appears twice).
Cf. A188674 (max part m appears m times), A001522 (max part m appears at least m times).
Cf. A001523 (max part appears any number of times).
Cf. A000009 (symmetric, max part m appears once; also symmetric, max part appears an odd number of times).
Cf. A035363 (symmetric, max part m appears twice; also symmetric, max part appears an even number of times).
Cf. A087897 (symmetric, max part m appears 3 times).
Cf. A027349 (symmetric, max part m appears m times), A189357 (symmetric, max part m appears at least m times).
Column k=3 of A247255.

N=66; x='x+O('x^N); Vec(sum(n=0,N, x^(3*n) / prod(k=1,n-1, 1-x^k )^2 ))

G.f.: sum(n>=0, x^(3*n) / prod(k=1..n-1, 1-x^k )^2 ); replace 3 by m to obtain g.f. for "... max part appears m times".

a(n) ~ Pi^2 * exp(2*Pi*sqrt(n/3)) / (16 * 3^(7/4) * n^(9/4)). - Vaclav Kotesovec, Oct 24 2018

A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 19 2021

nonn

First differs from A096825 at a(525) = 3, A096825(525) = 4.
First differs from A345926 at a(90) = 4, A345926(90) = 3.
The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime factors is also the reverse-alternating sum of reversed prime factors.
Also the number of distinct "sums of prime factors" of divisors d|n such that bigomega(d) = bigomega(n)/2 rounded up.

			The divisors of 525 with 2 prime factors are: 15, 21, 25, 35, with prime factors {3,5}, {3,7}, {5,5}, {5,7}, with distinct sums {8,10,12}, so a(525) = 3.

The half-length submultisets are counted by A114921.
Including all multisets of prime factors gives A305611(n) + 1.
The strict rounded version appears to be counted by A342343.
The version for prime indices instead of prime factors is A345926.
A000005 counts divisors, which add up to A000203.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A108917 counts knapsack partitions, ranked by A299702.
A276024 and A299701 count positive subset-sums of partitions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A334968 counts subsequence-sums of standard compositions.
Cf. A008549, A032443, A083399, A096825, A344609, A345957.

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Length[Union[Total/@Subsets[prifac[n],{Ceiling[PrimeOmega[n]/2]}]]],{n,100}]

from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
def A343943(n):
    fs = factorint(n)
    return len(set(sum(d) for d in multiset_combinations(fs,(sum(fs.values())+1)//2))) # Chai Wah Wu, Aug 23 2021

A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 7, 7, 13, 11, 11, 17, 13, 15, 25, 17, 17, 29, 19, 23, 35, 25, 23, 39, 29, 29, 45, 33, 29, 55, 31, 35, 55, 39, 43, 65, 37, 43, 65, 51, 41, 77, 43, 51, 85, 53, 47, 85, 53, 65, 87, 61, 53, 99, 67, 67, 97, 67, 59, 119, 61, 71, 113, 75, 79, 123, 67, 79, 117

Offset: 0

Gus Wiseman, Apr 02 2021

nonn

			The a(1) = 1 through a(9) = 13 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)  (1,8)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)  (2,6)  (2,7)
                          (3,2)  (4,2)    (3,4)  (3,5)  (3,6)
                          (4,1)  (5,1)    (4,3)  (5,3)  (4,5)
                                 (1,2,3)  (5,2)  (6,2)  (5,4)
                                 (3,2,1)  (6,1)  (7,1)  (6,3)
                                                        (7,2)
                                                        (8,1)
                                                        (1,3,5)
                                                        (2,3,4)
                                                        (4,3,2)
                                                        (5,3,1)

Wikipedia, Arithmetic progression.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Strict compositions in general are counted by A032020.
The unordered version is A049980.
The non-strict version is A175342.
A000203 adds up divisors.
A000726 counts partitions with alternating parts unequal.
A003242 counts anti-run compositions.
A224958 counts compositions with alternating parts unequal.
A342343 counts compositions with alternating parts strictly decreasing.
A342495 counts compositions with constant quotients.
A342527 counts compositions with alternating parts equal.
Cf. A000009, A001522, A002843, A049988, A062968, A070211, A114921, A325545, A325557, A342496, A342515, A342522.

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],SameQ@@Differences[#]&]],{n,0,30}]

a(n > 0) = A175342(n) - A000005(n) + 1.

a(n > 0) = 2*A049988(n) - 2*A000005(n) + 1 = 2*A049982(n) + 1.

A380108 Number of distinct partitions of length n binary strings into maximal constant substrings up to permutation.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 29, 48, 75, 118, 179, 272, 403, 596, 865, 1252, 1786, 2538, 3566, 4990, 6918, 9552, 13086, 17856, 24205, 32684, 43881, 58698, 78125, 103618, 136820, 180064, 236031, 308432, 401585, 521340, 674579, 870446, 1119786, 1436798, 1838405, 2346480, 2987204

Offset: 0

Yaroslav Deryavko, Jan 12 2025

nonn

Equivalently, a(n) is the number of partitions of n into parts of two kinds where the number of parts of each kind differ by at most one.

			For n = 3, the partitions are (000), (111), (00, 1), (0, 11), (0, 0, 1), (0, 1, 1).

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

Cf. A000712, A114921, A342528.

g:= (n, i, t)-> `if`(t>1+n, 0, `if`(n=0, 1, b(n, i, t))):
b:= proc(n, i, t) option remember; add(add(g(n-i*j,
      min(n-i*j, i-1), abs(t+2*h-j)), h=0..j), j=`if`(i=1, n, 0..n/i))
    end:
a:= n-> g(n$2, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, Jan 15 2025

g[n_, i_, t_] := If[t > 1+n, 0, If[n == 0, 1, b[n, i, t]]];
b[n_, i_, t_] := b[n, i, t] = Sum[Sum[g[n-i*j,
   Min[n-i*j, i-1], Abs[t+2*h-j]], {h, 0, j}], {j, If[i == 1, n, 0], n/i}];
a[n_] := g[n, n, 0];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 02 2025, after Alois P. Heinz *)

seq(n)={my(p=1/prod(k=1, n, 1-y*x^k + O(x*x^n))); Vec(sum(k=0, n\2, polcoef(p, k, y)*(2*polcoef(p, k+1, y) + polcoef(p, k, y))))} \\ Andrew Howroyd, Jan 12 2025

n = 0
while True:
    m = set()
    for i in range(2**n):
        t = bin(i)[2:]
        t = '0' * (n - len(t)) + t + '2'
        l = []
        s = 0
        for j in range(1, n + 1):
            if t[j] != t[j - 1]:
                l.append(t[s:j])
                s = j
        l.sort()
        l = tuple(l)
        m.add(l)
    print(len(m), end=' ')
    n += 1

G.f.: Sum_{k>=0} ([y^k] P(x,y))*([y^k] (1 + 2*y)*P(x,y)), where P(x,y) = Product_{k>=1} 1/(1 - y*x^k). - Andrew Howroyd, Jan 12 2025

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A226541 Number of unimodal compositions of n where the maximal part appears three times.

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A343943 Number of distinct possible alternating sums of permutations of the multiset of prime factors of n.

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A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.

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A380108 Number of distinct partitions of length n binary strings into maximal constant substrings up to permutation.

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