1,2

Row n gives integer exponents k such that in the expansion in powers of q of (q^(-1/24) * eta(t))^k the term q^n has a zero coefficient, where q = exp(2*Pi*i*t) and eta is the Dedekind eta function. Alternatively integer exponents k such that x^n in the expansion of f(-x)^k in powers of x has zero coefficient, where f() is a Ramanujan theta function.

Row n gives the real integer zeros of the polynomial defined in row n from A078521 the D'Arcais polynomials.

Some rows appear multiple times.

Jean-Pierre Serre proved that for even r the lacunarity of eta(x)^r is only reached iff r is from 2, 4, 6, 8, 10 and 26. Empirically the even positive numbers in this sequence are only from 2, 4, 6, 8, 10 and 26. Lehmer's conjecture on Ramanujan's tau function asserted the absence of 24.

Some odd numbers appear surprisingly late, for example the first occurrence of 5 is in row 1560, the first occurrence of 7 in row 28017, 15 appears first in row 53 however 9, 11 and 13 are not seen in a search up to 10^10 by Bernhard Heim.

Jean-Pierre Serre, Sur la lacunarité des puissances de eta, Glasgow Math. Journal, 27 (1985), 203-221.

Row n: Real integer roots of the polynomial P_{n}(x) = -(x/n)*Sum_{k=1..n} A000203(k)*P_{n-k}(x) for n > 0, with P_{0}(x) = 1. We can calculate this polynomial also directly per monomial: P_{n} = Sum_{k=1..n} (-1)^k*(x^k/k!)*Sum_{v_k from C_n} Product_{m=1..k} A000203(v[m])/v[m], where "v_k from C_n" means we take all elements of length k as vectors from the set of compositions of n. v[m] means we select individual components of the vector. See A066099 for an explanation of the term compositions.

0 = Sum_{k=2^n..2^(n+1)-1} ((-T(n, m))^A000120(k)/A000120(k)!)*Product_{j=1..A000120(k)} (A000203(A066099(k, j))/A066099(k, j)).

The Nekrasov-Okounkov hook length formula: Sum_{P from partitions of n} Product_{h from hooklengths in partition P} (1-((t+1)/h^2)) = 0 for t > 0 and from row n. See the example section for an explanation of hooklength h and P.

The triangle begins:

Row | List of numbers

1 | 0

2 | 0, 3

3 | 0, 1, 8

4 | 0, 1, 3, 14

5 | 0, 3, 6

6 | 0, 1, 10

7 | 0, 2, 3, 8

8 | 0, 1, 3, 6

9 | 0, 1, 3, 4, 14, 26

10 | 0, 1

11 | 0, 1, 2, 3, 8

12 | 0, 2, 3

...

Row 2 lists the number 3 because, if we consider k, k' and k~ to be different versions of k then we have 2, 2', 2~, 1+1', 1+1~, 1'+1~. The even partitions number 3 and the odd partitions number 3 too.

.

This table is a Ferrers diagram representing a partition of 14:

|2|1|_

|4|3|1| In each cell of the Ferrers diagram is the hook length entered.

|5|4|2|_ _ _ The hook length is the total number of cells in straight line

|9|8|6|3|2|1| above and to the right including the own cell.

(PARI) row(n) = my(p=factor(polcoef(eta(x+O(x^(n+1)))^y, n, x))[, 1], v=[]); for(k=1, #p, if(#(p[#p-k+1])==2, v=concat(v, -Vec(p[#p-k+1])[2]))); v;

Cf. A389567 (row length).

Cf. A000120, A000203, A066099, A078521, A286354, A319933.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), A020757 (m=6), A322430 (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

Sequence in context: A290776 A373086 A172249 * A208758 A320161 A373637

Adjacent sequences: A390310 A390311 A390312 * A390314 A390315 A390316

nonn,tabf

Thomas Scheuerle, Nov 01 2025

approved