Elena Lanina - cp's OEIS frontend

User: Elena Lanina

Elena Lanina has authored 2 sequences.

A334377 Irregular triangle read by rows: T(n,k) is the number of partitions of k into distinct parts p such that 2 <= p <= n.

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

Offset: 2

Victor Mishnyakov, Elena Lanina, Apr 25 2020

			Irregular triangle begins:
----------------------------------------------------------
n\k | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
----------------------------------------------------------
  2 | 1 0 1
  3 | 1 0 1 1 0 1
  4 | 1 0 1 1 1 1 1 1 0 1
  5 | 1 0 1 1 1 2 1 2 1 2  1  1  1  0  1
  6 | 1 0 1 1 1 2 2 2 2 3  2  3  2  2  2  2  1  1  1  0  1
  ...
For n = 4: T(4,3) = 1 because we have [3], G.f.=1+x^2+x^3+x^4+x^5+x^6+x^7+x^9;
For n = 5: T(5,5) = 2 because we have [5] and [3,2].
G.f. is 1+x^2+x^3+x^4+2x^5+x^6+2x^7+x^8+2x^9+x^10+x^11+x^12+x^14.

Cf. A000009, A000041, A025147, A087897, A334305.

trow[n_] := CoefficientList[Product[(1 + x^i), {i, 2, n}], x]; nmax = 10; Table[trow[n], {n, 2, nmax}] // Flatten

G.f. for row n: Product_{i=2..n} (1+x^i), n >= 2.

A334305 a(n) is the number of partitions of n of the form [k,k,b(1),b(2),...], where k>=b(1)>b(2)>...>=2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 2, 3, 2, 4, 4, 4, 6, 6, 7, 8, 10, 10, 13, 14, 16, 18, 22, 22, 28, 30, 34, 39, 44, 48, 56, 62, 69, 78, 88, 96, 110, 122, 134, 152, 168, 186, 208, 231, 254, 284, 314, 346, 384, 425, 466, 518, 570, 626, 692, 762, 834, 922, 1010

Offset: 0

Victor Mishnyakov, Elena Lanina, Apr 22 2020

a(n)>0 if n>=2k>=4.

			a(4)=1 because we have [2,2]; a(6)=2 because we have [2,2,2] and [3,3].
G.f.= x^4+2x^6+2x^8+x^9+2x^10+2x^11+3x^12+2x^13+...

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

Cf. A000009, A000041, A025147, A087897.

nmax = 100; CoefficientList[Series[Sum[x^(2 k) Product[(1 + x^i), {i, 2, k}], {k, 2, nmax/2}], {x, 0, nmax}], x]
Flatten[{{0, 0, 0}, Table[PartitionsQ[n + 3] - 2*(-1)^n + 2*Sum[(-1)^k * PartitionsQ[n - k + 3], {k, 1, n - 2}], {n, 3, 70}]}] (* Vaclav Kotesovec, Apr 24 2020 *)

G.f.: Sum_{k>=2} x^(2k) Product_{i=2..k} (1+x^i).
From Vaclav Kotesovec, Apr 24 2020: (Start)
For n>=3, a(n) + a(n+1) = A087897(n+4).
a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (16 * 3^(3/4) * n^(5/4)). (End)

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User: Elena Lanina

A334377 Irregular triangle read by rows: T(n,k) is the number of partitions of k into distinct parts p such that 2 <= p <= n.

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