A272689 Irregular triangle read by rows: row n (n >= 1) gives entries of the g-vector for [n]!.

1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 2, 1, 4, 9, 15, 20, 22, 19, 11, 1, 5, 14, 29, 49, 71, 90, 100, 96, 76, 42, 1, 6, 20, 49, 98, 169, 259, 359, 454, 525, 553, 524, 433, 286, 100, 1, 7, 27, 76, 174, 343, 602, 961, 1415, 1939, 2486, 2990, 3374, 3562, 3493, 3134, 2489, 1602, 553

Offset

1,6

Links

Table of n, a(n) for n=1..67.

Charles Brittenham, Andrew T. Carroll, T. Kyle Petersen, and Connor Thomas, Unimodality via Alternating Gamma Vectors, Electron. J. Combin., Volume 23, Issue 2 (2016), Paper #P2.40. See Table 1 p. 19.

Example

Triangle begins:
  1;
  1;
  1, 1;
  1, 2,  2,  1;
  1, 3,  5,  6,  5,   2;
  1, 4,  9, 15, 20,  22,  19,  11;
  1, 5, 14, 29, 49,  71,  90, 100,  96,  76,  42;
  1, 6, 20, 49, 98, 169, 259, 359, 454, 525, 553, 524, 433, 286, 100;
  ...

Prog

(SageMath)
@cached_function
def g(n, i):
    if n == 1: return 1 if i == 0 else 0
    return sum(g(n-1, j) for j in (i-(n-1) .. i if i <= ceil(binomial(n-1, 2)/2) else binomial(n, 2)-i-(n-1)))
for n in (1..9):
    print([g(n, k) for k in (0..binomial(n, 2)//2)]) # Andrei Zabolotskii, Sep 23 2025

Crossrefs

Row sums seem to be A000140.

Cf. A008302.

Sequence in context: A076037 A215563 A076263 * A274887 A008302 A131791

Adjacent sequences: A272686 A272687 A272688 * A272690 A272691 A272692

Keyword

nonn,tabf

Author

N. J. A. Sloane, May 27 2016

Extensions

Row 9 from Andrei Zabolotskii, Sep 23 2025

Status

approved