A054242 -id:A054242 - cp's OEIS frontend

A268347 Number of partitions of (4, n) into a sum of distinct pairs.

2, 7, 14, 27, 46, 74, 116, 174, 254, 363, 510, 703, 957, 1285, 1706, 2244, 2924, 3777, 4844, 6168, 7802, 9813, 12272, 15267, 18902, 23295, 28584, 34935, 42532, 51592, 62369, 75150, 90265, 108102, 129094, 153743, 182627, 216395, 255792, 301672, 354994, 416851

Offset: 0

Vaclav Kotesovec, Feb 02 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Column 4 of A054242.

max=50; col=4; s1=Series[Product[(1+x^(n-k)*y^k), {n, 1, max+2}, {k, 0, n}], {y, 0, col}]//Normal; s2=Series[s1, {x, 0, max+1}]; a[n_]:=SeriesCoefficient[s2, {x, 0, n}, {y, 0, col}]; Table[a[n], {n, 0, max}] (* after Jean-François Alcover *)
nmax = 50; CoefficientList[Series[((2 + 3*x - x^3 - 4*x^4 - 2*x^5 + x^6 + x^7 + 2*x^8 - x^9) / ((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)))*Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 3^(3/4) * n^(5/4) * exp(Pi*sqrt(n/3)) / (2*Pi^4).

A268348 Number of partitions of (5, n) into a sum of distinct pairs.

Original entry on oeis.org

3, 10, 21, 42, 74, 123, 197, 303, 452, 659, 943, 1323, 1830, 2496, 3363, 4485, 5922, 7748, 10058, 12958, 16578, 21077, 26637, 33476, 41855, 52077, 64496, 79536, 97683, 119505, 145671, 176948, 214225, 258542, 311085, 373227, 446553, 532873, 634265, 753118

Offset: 0

Vaclav Kotesovec, Feb 02 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Column 5 of A054242.

max=50; col=5; s1=Series[Product[(1+x^(n-k)*y^k), {n, 1, max+2}, {k, 0, n}], {y, 0, col}]//Normal; s2=Series[s1, {x, 0, max+1}]; a[n_]:=SeriesCoefficient[s2, {x, 0, n}, {y, 0, col}]; Table[a[n], {n, 0, max}] (* after Jean-François Alcover *)
nmax = 50; CoefficientList[Series[((3 + 4*x + x^2 - 4*x^4 - 5*x^5 - 4*x^6 + 2*x^8 + 3*x^9 + 3*x^10 - x^12 - 2*x^13 + x^14) / ((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^5)))*Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 3^(5/4) * n^(7/4) * exp(Pi*sqrt(n/3)) / (5*Pi^5).

A378126 Array read by antidiagonals: T(n, m) is the maximal size of partitions of (n, m) into sums of distinct pairs of nonnegative integers, excluding (0, 0).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 5, 4, 4, 4, 3, 3, 4, 5, 5, 5, 5, 5, 5, 4, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 5, 5, 5, 4, 4, 5, 5, 6, 6, 6, 6

Offset: 0

Jimin Park, Nov 17 2024

A378379(n) is the least number x such that T(x, x) >= n, and as the growth rate of A378379(n) is Theta(n^(3/2)), the growth rate of T(n, m) is O((n+m)^(2/3)).

			Table begins:
  0 1 1 2 2 2 3 3 3 3 ...
  1 2 2 3 3 3 4 4 4 4 ...
  1 2 3 3 4 4 4 5 5 5 ...
  2 3 3 4 4 4 5 5 5 6 ...
  2 3 4 4 5 5 5 6 6 6 ...
  2 3 4 4 5 5 6 6 6 7 ...
  3 4 4 5 5 6 6 6 7 7 ...
  3 4 5 5 6 6 6 7 7 7 ...
  3 4 5 5 6 6 7 7 7 8 ...
  3 4 5 6 6 7 7 7 8 8 ...
  ...
T(9, 5) = 7, as (9, 5) can be expressed as the sum (0, 1) + (0, 2) + (1, 0) + (1, 1) + (2, 0) + (2, 1) + (3, 0), which is the longest for (9, 5).

Jimin Park, Table of n, a(n) for n = 0..1000

Maximal size among partitions considered by A054242 and A201377.
The first row is T(n, 0) = A003056(n).
Cf. A086435, A378379.

import functools
@functools.cache
def A378126(n: int, m: int, t: tuple[int, int] = (0, 0)) -> int:
  if (n, m) <= t: return 0
  v = 1
  for nn in range(t[0], n//2+1):
    for nm in range(m+1):
      if (nn, nm) <= t: continue
      rn, rm = n-nn, m-nm
      if (rn, rm) <= (nn, nm): continue
      nv = 1 + A378126(rn, rm, (nn, nm))
      if nv > v: v = nv
  return v

T(n, m) = T(m, n).
T(n, m) >= T(n, 0) + T(0, m).
T(n, m) = A086435(p^n * q^m) for any distinct primes p and q.
T(n, 0) = A003056(n).
T(n, 1) = T(n, 0) + 1.
T(n, 2) = T(n-1, 0) + 2, for n >= 1.
T(n, m) = O((n+m)^(2/3)).

A378379 Minimal x such that there is a partition of (x, x) into sums of distinct pairs of nonnegative integers with size at least n, excluding (0, 0).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 9, 10, 12, 14, 16, 18, 20, 23, 25, 28, 30, 33, 35, 38, 41, 44, 47, 50, 53, 56, 60, 63, 67, 70, 74, 77, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 129, 134, 138, 143, 147, 152, 156, 161, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220

Offset: 1

Jimin Park, Nov 24 2024

nonn

For (n, n), there is at least one maximal partition P that's symmetric: (x, y) in P <=> (y, x) in P. This can be proven by manipulating integer sequences c(i) (i >= 1) such that 0 <= c(i) <= i+1 for all i and Sum_{i > 0} i*c(i) = 2n, which correspond to partitions P of (n, n) with size |P| = Sum_{i > 0} c(i), where c(i) is equal to number of (x, y) in P such that x+y = i.

			For n = 8, a(n) = 9, as (9, 9) can be expressed as the sum (0, 1) + (0, 2) + (0, 3) + (1, 0) + (2, 0) + (3, 0) + (1, 2) + (2, 1), but the longest sum for (8, 8) has 7 pairs.

Maximal size among partitions considered by A054242 and A201377.
Minimal x such that A378126(x, x) >= n.
Cf. A086435.

import math
def A378379(n: int) -> int:
  l = (math.isqrt(1+8*n)-1)//2 # l = A003056(n), min. possible largest pair norm
  r = n - (l-1)*(l+2)//2 # r = n - A000096(l-1), number of pairs with norm l
  return ((l-1)*l*(l+1)//3 + l*r + 1)//2 # ceil((A007290(l+1) + l*r) / 2)

a(n*(n+3)/2) = n*(n+1)*(n+2)/6.

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A268347 Number of partitions of (4, n) into a sum of distinct pairs.

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A268348 Number of partitions of (5, n) into a sum of distinct pairs.

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A378126 Array read by antidiagonals: T(n, m) is the maximal size of partitions of (n, m) into sums of distinct pairs of nonnegative integers, excluding (0, 0).

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A378379 Minimal x such that there is a partition of (x, x) into sums of distinct pairs of nonnegative integers with size at least n, excluding (0, 0).

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