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

2,25

Links

Table of n, a(n) for n=2..84.

Formula

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A129174 A129175 A373382 * A063053 A063050 A330098

Adjacent sequences: A334374 A334375 A334376 * A334378 A334379 A334380

Keyword

tabf,nonn,easy

Author

Victor Mishnyakov, Elena Lanina, Apr 25 2020

Status

approved